(MOST OF THE EQUIPMENT IMAGES ARE FROM B&H PHOTO... Thanks B&H)
STUDIO LIGHTING... What is the best way to go.... (or Best way to start)
Experience is the best way to go. Experience has always been my best teacher. I've taken classes and seminars but when it comes down to it. Trial and error is what makes you grow.
First thing you need is a good portrait lens. I use a 28-70 in the studio most times and i have a 50MM that i like to use at times.
I have 3 main setups for most studio shoots. Everything is pretty consistent except for the light placement and power settings.
SETUP 1 See Diagram
1 is the backdrop or background.
2 is the hair light. It is set at the same strength as the fill strobe.
3 is my main Strobe.
4 is the subject and
5 is the fill strobe.
The subject is about 4 feet from the background. The hair light is mid way in between the subject and the background. I use a 20 degree grid on the hair light. That directs light only to a small place on the back of the subject. The hair light is about 2 feet above the subject and aiming down at them. In general your strobes #3 & #5 should be at the same height as the subjects face or a little higher. ( a few inches). As for power setting. The fill strobe should be set at 1/2 the power of the main. And the hair light at the same as the fill. This variation in power settings gives you depth in the image not to be confused with Depth of Field. You can vary the power setting on the fill strobe to less for more depth or more for less depth. LOL shoot these at an f-stop of about 2.8 to 4.0 for a shallow depth of field. I stay back from the subject and zoom in. This is a Classic Portrait set-up.


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TIP
When photographing small children. Turn your strobes power down as much as possible. The strobes flashes will tire a little one out fast. This is actually ideal. Because if your f-stop is at 5.6 then you can lower it to 4.0 or 2.8 giving you a shallower depth of field which is ideal for most portraits.
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I had a question from someone about the hair light.
You can attach it to the ceiling if possible. But ideally it would be best to use a light stand and boom. It makes it very versatile. Amvona has them at decent price. Make sure you can attach your light to the boom before you buy it. They aren't all universal. One add note i let my hair light fire via the optical slave mode.

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Here is another set up i use often.
I call it "High Key"
Equipment...
2 Strobes with umbrellas (Not shoot thru)
1 soft box
1 White background
In this one i have the subject exposed correctly and the background is blown out or nearly blown out.
Make sure your subject stays in front of the two umbrella lights or you will get that light spilling over.
You will get some bounce light off the white background that will light the edges of your subject.
When placing your umbrellas aim one slightly upwards and the other slightly downwards. This will help cover the whole background with light.
Your softbox should be at eye level with the subject or just a bit higher. In this set up you can get some really nice catch lights in the eyes.
There are some photographers on this forum that do some excellent High Key photos. I would love to see their setups.
If you are doing a full body shot and see the feet or the subject is sitting on the floor. A nice addition is that white panel board you can get from the lumber yard. Its somewhat reflective. Make for a nice image. My backdrop is actually white canvas. Its easier to shoot since there isnt wrinkles. White paper works well too.


Nice "ONE LIGHT SETUP"
In this setup you can achieve a very nice looking portrait with one light. And a few modifiers.
Backdrop, any color
1 Main light with shoot thru umbrella.
Reflector (modifier) (can be just about anything that is very reflective.) That white panel board, you'll need to get it to stand up somehow. I use a popup reflector that i hang from a light stand
You will need to place the reflector about 1 1/2 to 2 feet from the subject and see that alot of light from your umbrella is hitting it.
Place your light about 4 feet from the subject and then a slightly wider angle than the normal set-up. I place the umbrella slighty higher than eye level about 6 inches and aim it down to the eye. You will get a front hair light from this.
this setup will natually make a shadow so keep your subject far enough away from the background so the shadow is out of frame but close enough that you get some light from the strobe lighting the background.

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An explanation of terms is an excellent idea Karrie.
1. Strobes - This is the most common name used others are monolight, flashes. A strobe is a fancy flash. In fact some use flashes that can be mounted on the camera as strobes. It consists of a power supply; either a plug for the wall or a battery pack. The strobe is itself is a bunch of capcitors and electronics. The Strobe bulb; is the same sort of thing that most cameras have just more powerful. A modeling lamp; this is a regular bulb around 100 watt that gives the photographer some idea where the lights are aimed. The modeling lamp or light stays on. While the strobe bulb only fires or flashes when triggered. Strobes must be triggered in some way. They arent mounted to your hotshoe so you need a cord called a PC snyc that connects the strobe to the camera. You can also use a radio trigger. This is basically a wireless pc snyc that that has 2 parts one part you attach to the strobe the other attaches to the camera. And it use radio frequency to fire the strobe. You can also use ab IR trigger; these use infared light to fire the strobes the same thing that works the remote on your TV. But as with the TV, it needs to see the strobe. There is the optical firing method. most strobes have an optical eye that will see a flash and it will fire itself in conjunction.

Strobes have different terms associated with them depending how they are used.
Master- this is the one that is connected to the camera via any of the triggering methods i mentioned
Slave - all other strobes are called slaves that are being used for the shoot. They fire because the master fires. Alot of times the Master is connected via radio or pc snyc and the others are triggered via optical. You would not usually fire the master strobe with an opitcal trigger because that would mean you'd have a flash on camera and that wouldnt be a very nice photo.
Hair light. - A slave strobe that fires behind and above the subject and possible a few inches to the left or right of the subject to give the hair a nice glow. Often set at the same power as the fill light. or at least 1/2 or 1/4 power of the main light.
Fill Light. This is the slave strobe that is used to add fill or reduce shadow. Its usually 1/2 the power of the main light.
Main Light. This is usually the master strobe that is connected to your camera and is porducing most of the light. For exposure i take a reading from the main light and adjust it down 1 stop to compensate for the fill. This is what i do its not a hard and fast rule.
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Modifiers is generic term for just about everything else associated with your strobes. They modify how the light affects the subject. Most modifiers soften light or direct light in someway.
Umbrella- When most people mention umbrella they are speaking of a bounce umbrella. One where there is a black coating on the outside and silver or white coating on the inside. The strobe faces the umbrella. And the light bounces off of the silver coating difusing or soften the light. It also helps it cover more of your subject.

Shoot Through Umbrella- Like the name suggests, the light shoots (passes) thru the umbrella. This umbrella is made of a white thin material. Your subject will see the back of the umbrella as opposed to a bounce umbrella where the see the inside.

Brollybox - Another very useful modifer is a combination of the bounce and shoot thru umbrella. Shaped like an umbrella it looks just like a shoot thru umbrella and is placed on the strobe the same way. However the is a black and silver reflector that covers the opening away from the subject that bounce residual light back thru the umbrella part. (see the image)

Softbox- very similair to the brolly box with 2 very important differences. The Soft box has a flat front. And inside the soft box there is primary difuser that is the first thing the light passes thru the secondary difuser on the outside. The primary difuser allows some light to bounce back to the metallic coated interior. And then back out again. Again a diffuser in this case is a semi-transparent cloth.This actually makes the light bounce around in the softbox quite abit giving it a very very nice soft glow in your images. Soft boxes come alot of shapes and size 20" x 20" up to 40" x 72" maybe larger. You can use a strip light softbox for hair which is great for larger groups. You can even get a soft box with a grid but more on that later. They can be square,rectangle, or Octagon. A softbox is my prefered modfier in my studio.

Snoots, grids and barn doors- while the umbrella and softbox expand the light to a large area. Snoots and grids and barndoors direct the light to a smaller area. or specific area. Snoots are tubular shaped attachments that you place on the end of the strobe to direct the light into one small place. I have one and i dont think i ever used it. Not to say one never will. Grids are a honeycomb type attachments for your strobes. According to the size of the holes in the grid is what makes them vary from one to another. They go by degree. I use a 20 degree grid on my hair light so it directs the light into one small area. But not small as a snoot. Barndoors are another attachment for the strobe. It is used to box the light into one certain area. Say you strobe is aiming somewhat towards the camera for one reason or another. You can use a barndoor to keep the strobes light from hitting the lens directly.
Grids 10 deg - 40 deg

Barndoors

Snoot on Strobe

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There are 4 basic light styles I'd like to note.
The first 2 go hand in hand.
Broad Lighting and Short Lighting.
The easiest way to understand and remember these 2 lighting techniques is this.
When your subject is facing your main light. ie a BROAD amount of light is hitting them then this is BROAD LIGHTING.
When your subject is facing your fill light or reflector. ie a small or SHORT amount of light is hitting them, then this is SHORT LIGHTING.
The other 2 types of lighting also go hand in hand.
Butterflying Lighting and Rembrandt lighting.
Butterfly lighting is a bit more complicated than broad and short lighting. Your main light is postition directly in front of the subject and at least 2 feet higher than the subject aiming down at them.
See this side view example

And this is the top view

Notice the the light is high and low in this lighting setup. The main light is high aiming down and the fill or reflector is low aiming up. This look is flattering for women more so than men. The name comes from the butterfly shadow under nose and lip. The fill is optional. I dont care from the strong shadow so that why i included it in the example. Test it for yourself and see which you like.
Here is the Rembrandt light set-up. I cant say ive ever used this setup. I just dont care for it. Its is a combo of short light and butterfly.
Here is a side view (It looks the same from the side as does the butterfly.)

The top view is completely different.

Notice the main light is one the short side. and the fill is on the broad side. Hardcore Rembrandt lighting wouldnt use a fill. But there again i dont care for the harsh shadow. The Rembrandt lighting is named after Rembrandt the Dutch painter. He used this light scheme in his paintings and may have developed it, this im not sure of. The key look of a rembrandt lighted image is the triangle shape under the eye. Where the shadow of the nose and cheek make a shadowed area. These 2 lighting setups show great depth and character.
Photo Example of rembrandt light. Click this link below. This is an excellent example of rembrant light. notice the lighted triangle on her left cheek (Right one facing us) This is key in rembrant light and it is also shot short where the subject is facing away from the main light.
http://www.flickr.com/photos/bertavillanueva/2274893302/