Hey there Doafan,
You'll definitely want to get your hands on a power and AV cable if you need to test them. Since the power supply is built in, you should be able to just use one of those two-hole power cords like the Dreamcast uses (I think any similar cord will work, I can't think of what they are actually called though). You'll have to buy the AV cables, or borrow some from a friend.
If they turn on, try out a game. Make sure to play it for at least like 15 minutes, because one problem you may run into (depending upon the age of the 'box) is the disc drive. You can look this up online, but I believe there were two or three different disc drives they used throughout the Xbox's life span; if you have one of the older ones, it has a disc drive (Thompson?) which is especially prone to failure (i.e. you can play for a while, but at some point around 10 or 15 minutes in it'll say there's a disc error, because the drive fails to function). These are kind of expensive to buy on their own; you can buy another Xbox for the $30 or $40 you'll pay to get a new one.
If you want, you can open them up by taking the six screws out of the bottom, removing the shield over the Disc drive (left side, if console is sitting in front of you in a normal position), and looking at the drive; again, if you google search you can find pictures that'll clearly show you how to identify which drive it is. (Edit: This is very easy to do, you just need the right screwdriver.)
Other than that, I'm not sure if Xbox's have any other big problems. I have my original, and aside from the disc drive failing I haven't had any issues. It's a great system (games look better on it than the PS2), and since it doesn't have the perceived value that Nintendo consoles have, you can get games cheaper on it than the GameCube equivalent.
Also, I'd highly recommend playing Metal Arms: Glitch in the System (you can find it pretty cheap) if you like 3rd person shooters with a nice amount of content. I'm not sure how well it holds up nowadays, but I really enjoyed it back in the day.