I'm assuming I could just experiment with how I want to arrange them as far as using one for key light and the other for either fill or backlight, right?
Yes, absolutely.
Also, I read somewhere that you're better off not using a backlight on small children. How do you guys feel about that?
I wouldn't say 'better off'...but I think the issue is that small children don't sit still...and they may not be in the right position for the backlight to be effective. But there is no reason not to try it if you think it will look good.
Umbrellas or softboxes? Does it really matter if people are my subjects? I've read that umbrellas are bad if you're photographing shiny objects like bottles and glass
As mentioned, umbrellas tend to spread the light all over, which isn't a big deal if you are shooting on a light background, but if you are trying to shoot on a darker background, especially in a small space, then the light may bounce all around and put too much on your backdrop. A softbox keeps the light to just one direction, which helps you to better control the light...however, they are a lot more expensive than umbrellas and most of them are a lot harder to set up and take down than an umbrella.
They both do the same basic job though, which is to soften the light.
If I bought one of those starter kits, what else do I need to get it functional?
The first thing is that you need a way to trigger the lights. Most strobes will come with a cord...but your camera may or may not have a sync port to plug the cord into. If your camera doesn't have the PC flash sync port, then you will need a hotshoe adapter, something like
THIS. However, many of us have found that it's easier to work with a wireless trigger system. There are several options, ranging from $40 to several hundred. Check out
THIS one.
It won't be cheap, but a really good tool to help you out with your studio lighting, will be a flash meter. I recommend
THIS one or
THIS one. These will allow you to precisely measure your lighting, which then allows you to set up lighting ratios (main/fill/back) and find the proper exposure settings for the camera.