I'm by no means an expert, but I've taken a couple classes so I'll answer the best I can. For much better information, I'd suggest going over to a forum like http://www.offroadfabnet.com/
1. What of steel to use for bumpers?
Carbon Steel is the standard I believe for most stuff. If you go to your local metal job and ask for plate, it'll probably be carbon (I think, someone correct me if I'm wrong). But to answer your question, it depends on what your looking to. I mean if you want to build a PreRunner tube bumper, go with something like 1.75" (.120 or thicker) DOM Tube. Probably need a bender of some sort to make it look decent though.
If you just want to build a big honkin' no-one-will-ever-damage-this bumper, go with plate. 1/8" will probably work. 1/4" would be beefy. 3/8" would be really strong. It'll be heavy as hell if it's normal size, but we need to be more specific to know what your looking for.
I mean you could even go with Chromoly stuff or something if you just had money to spend, but that kind of stuff would better be spent on frames in my opinion and wouldn't be worth dropping the $$$ on for a bumper.
2. What metals weld (stick) together?
All kinds? It doesn't really make sense. You can weld anything. Hell you can weld plastic (it's not really the same process but they call it welding). Obviously some metals like Chromoly and Aluminum are a bit more picky and to really weld them well, you need to be good at TIG welding. But any type of steel you can MIG. And with the right setup, you can MIG aluminum and Chromoly but you have to be damn good and know what your doing, so I wouldn't recommend it if your new to welding. Stick with standard Carbon Steel with a Wire Feed MIG.
If you don't know, TIG stands for Tungsten Inert Gas and MIG stands for Metal Inert Gas. Sometimes they are referred to as GTAW (Gas Tungsten Arc Welding) or GMAW (Gas Metal Arc Welding). MIG is a much easier process to learn, is very quick, but has less applications to due to inability to have fine control over the weld. TIG on the other hand is much harder to get good at, takes much longer, but you can finely control the puddle which allows for much higher quality welds. For more information, I'd just hit up google for awhile.
3. What kind of welders to use?
If your first learning, pick up a MIG. 220V Welders will get you way better quality welds then a Home Depot special. But those Home Depot welders definitely have there place. If you are doing light duty welding, then a small 120V will work fine. Especially considering the price tag on a good 220V welder.
4. How thick of tube for bumpers??
It depends on application. I mean go measure the bumper that's on your truck right now. That metal is probably only 1/8" thick if that. .120 (Tube standard thickness is done in decimal form) would work. That's pretty much the standard. Rollcages get made out of .120 so it'll work for bumpers. If your worried about hitting a lot of stuff, go with something a little thicker. It'll weigh more, but you'll repair it less.
5. Does it have to be bare metal for a good weld?
If you want your welds to be really clean and look professional, yes! Clean your metal. Make sure that all those oils and rust and other crap is all cleaned off. It'll make your welds so much better. This was the first lesson I learned in my classes and it's one I won't forget. If you have spare metal lying around, try welding a rusty piece of steel, followed by a nice clean freshly ground piece of steel. World of difference in the quality of your puddle and the ability to control it.