Welcome to the board!
Here's a couple of thoughts for you...
For random do it yourself stuff, a good little 120V mig welder with the gas option will cause you the least amount of headaches. I currently have a variety of welders and have used a variety in the past. My Century 110 with infinite heat adjustment, wire speed and a good sized bottle of gas has served me well. I also have one that only uses the flux core. The gas is certainly nice and produces nicer welds.
The stick welder is nice for heaver steel and structural welding.
Basically, for about 1/8" and less, the mig is the way to go. If you are consistently going to weld heavier stuff, then a stick welder may suit you better.
For your $1000 budget, you would definitely be able to get a nice 120v mig welder, helmet, gloves, etc. I personally would look for a used one to bring the price down, but sometimes it is satisfying to buy new to eliminate a variable..... If you find some deals, you may even be able to get one of each on your $1000 budget. A number of years ago, I think I paid about $500 for the century with gas from the welding store--that included a gas bottle. We ended up trading it in for a larger bottle as we only got the little one. A couple of years ago I picked up the flux core Century for $40 on craigslist. A year ago, I picked up an AC/DC lincoln stick welder for $180. After that, it is just money on consumables......
Check out the Hobart line at Tractor supply stores. Lincoln and Miller are always good options--I have had some of each....
Once you have it, practice, practice, practice...browse the Fabrication section on here. there are already some excellent threads on techniques, etc.
Best of luck!