All of the advice here on equipment needs is excellent! But I'd like to offer you some other advice that it took me a while to learn. Now that I know about these things, we're making beer that is so much better than when we started brewing back in March.
You'll learn fast that mashing and making wort if not all that complicated. The more challenging part is what you should do once your boil is through, and you flame out your kettle.
The three things that most raised the quality of our beer (in a dramatic way) were the following:
1. Always make and pitch a yeast starter. Lots of instructions online for the best way to do that. And you really should consider investing in a stir plate and flask to grow your starters. It's the best way, and a 2L stir plate will only set you back about $45. Well worth the investment! Even without a stir plate, though, you can still gently agitate your flask as a way to get it oxygenated. But pitching the right amount of yeast cells is really important!
2. Make sure you pitch your yeast at the right temperature. I feel that for ale yeasts, your post-boil wort temp should be somewhere in the mid 60s F. Yeast manufacturers will tell you the optimum temp for ale yeasts in 70-75, but I feel that's too warm to start, since the fermentation process will naturally raise the temp inside your bucket or carboy.
3. Find a way to control your fermentation temperature. This has been the most-important lesson for us. If your ambient room temp, where you keep your bucket or carboy is 70 degrees, chances are it's between 75-80 inside your fermenter, which is fine for yeast growth, but risks you creating off flavors, and unwanted esters. The last two beers we did (an American stout and an IPA) fermented even and clean in about 7 days because we built a fermentation chamber where we could keep the temp outside of the bucket at a constant 63 degrees.
There are lots of low-tech methods (including swamp coolers) that you can learn how to make as a way to better control temp during fermentation. We were fortunate to have an old fridge in the garage, which we added a temperature controller to, but there are less expensive ways to get this done. Controlling temp is also the best way to make beer that is consistent.
Good luck, and welcome to the hobby. If you're like us, you'll get obsessed too!