Actually, all three of the power saws you mention, are really good at one or two things, but not great for everything.
For example, the chop/cut-off/miter saw, whatever you want to call it is great for cutting nice square ends or perfect angle cuts, but depending on the size of the saw, you are limted to the width of the piece you can cut. Table saws are great for ripping long lengths into specific sizes, and will handle large pieces of stock. A Band saw is probably the most versatile when it comes to cutting the small pieces we use in the model business.
Personally, if I could only have one, I think I would go with the chop/miter saw. This saw will cut very specific lengths, and angle cuts for the narrow width stuff, and will cut a perfectly square end very easily. This saw is also invaluable around the house, especially for any projects that require trimming. (doors/windows/baseboards etc.) I actually have 2 of them, of which neither is really a "High End" model. Kinda' wish I had one that had the laser guide, and a slider, that allows you to cut wider stock. The ones I have will only cut about 6 inch material. Any wider, you have to switch the material around, and cut from the other side, and sometimes it's hard to truly get a nice clean total cut this way.
Usually, if you need longer length material ripped to specific sizes, your local lumber yard will have a service where they will cut it for you. Personally, although I have a table saw and do use it on occasion, I don't particularly like using it if I don't have to.....they scare me a bit. You certainly should have 2 people on hand if you are cutting long lengths of anything.
I do also have a small bench top band saw, which I only picked up recently at a yard sale for $50, and have used it although sparingly. They are a nice tool to have, and certainly there are instances where that would be the best choice, although I do feel the chop/miter saw, would be used more frequently.
Isn't it funny, in the three answers you get, none vote for the same tool. I guess it will pretty much come down to your own decision, and the one you think you will use the most. Or better yet, save up and get them all. In reality, today, there are a lot of reasonably priced power saws of all types available, unless of course you want "Contractor Grade" equipment, which is probably not necessary for what you will be using it for.
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