I have a degree in computer science and write software for a hospital, before you ignore the wall-o-text
If you don't know how to assemble a computer, then don't try your first one by yourself. Your best off either buying a prebuilt machine, or having someone you know put it together for you (or with you so you can learn). I've seen a lot of "experts" screw up brand new computers by doing simple things.. like putting in the wrong motherboard screws, and watching it fry itself when it gets turned on. Just make sure you have someone that knows what you're doing if you are buying separate parts. As to what to get.. depends on your budget. If your gateway is 10 years old i would keep everything there and use it as a back up computer.. to download files you don't trust, music, etc.. i have a junker computer for that. But anyway.. for your gaming experience, you need a fast processor/motherboard, a good video card, and a good amount of RAM.
I would recommend anything 3.0 ghz or higher for the processor
1GB + video card
4G of Ram +.. depending on your operating system.. Windows XP is capped at like 3.5 or something but windows 7 you can get 8 or more i am pretty sure.
I would get a new monitor too, i play on a 22" monitor i got free from work, but dont sit too close like me because it will burn your eyes
you can get fairly cheep parts and/or computers from
www.newegg.com
www.mwave.com (doesn't charge tax)
they are fairly popular and fairly cheap.. and you are probably going to get a cheaper computer online rather than going to best buy, just make sure you have someone double check what you're getting. i spent 500 in parts not including the monitor and i haven't experienced the word.. wait.. what was it.. oh yeah "lag" since.
Good luck