Chop Saw
You need to cut your steel stock. Imagine welding a triangular bracket: two pieces welded at a 90 degree angle and one cross-brace connected to them at a 45 degree angle. If you're just welding flat stock, you don't need to miter the corners; you can just cut your stock to length. But, if you're welding channel or square-tube, you want a metal chop saw or cut-off saw capable of making an angle cut. If you don't have a chop saw, it can also be done with an angle grinder and cut-off wheel.
Jig or Corner Vise
Jigs can be a critical part of welding brackets or any other projects -- especially if you're making a lot of them and uniformity is important. The jig serves two functions. One is to hold your pieces in place so you can optimize your welding position, and the other is a reference to make sure you haven't warped or wracked your work. For brackets, you might use a corner vise like a picture-framing vise, or custom weld a jig to be used with C-clamps.
Welder
Any kind of welder is capable of doing this job from a stick welder to wire-feed to TIG. For the beginning welder, wire-feed metal inert gas or MIG welders with gas shielding is probably the easiest. The wire-feed feature eliminates the tricky striking action required to begin the arc of an arc welder. And gas shielding tends to be more forgiving than than flux -- especially if it takes you a time or two to get started and your first efforts leave a bit of carbon on your material.
Angle Grinder
An angle grinder is a very versatile and frequently used welder's tool. In fact, it's not at all unreasonable to have two or three angle grinders setup for different purposes: perhaps one with a flap sanding wheel, one with a cut-off wheel, and one with a grinding disc. When you're done with your welding, you can clean it up a little or a lot with any of these three angle grinder attachments.