my-name-is-mario:

Things to consider: Small budget, I’m just a teenager, it’ll take me some time to get all these things, I don’t know much on building computers.

Now I want to start with 2 dual core processors because quad just seems too much
Still not sure what I’m going to do about fans or vents
I need help choosing a motherboard
I want RAM, 3-5 GB should be fine for gaming (right?)
Major help with power supply
I can find a case some other time
Not sure what I’m doing with drives

Definitely wait for huge sales. Sure, if you want to buy things—like now—definitely don’t wait for the next Black Friday or Christmas sales, but maybe look for sales during upcoming holidays in the next few months. Plan out what you want and just patiently purchase the components you need.

Processor/Motherboard: Don’t skimp on the processor. Microcenter has a great a deal on pretty good motherboards that come with a free or discounted AMD processor that starts from dual-core, tri-core, quad-core and I think 6-core (I’m not sure about the 6-core). I am an AMD fan, but at this moment Intel has the best processors for your buck. Definitely look at Microcenter for that deal. 

Tower/Case: Google quiet/silent pc. You will get a good website that offers options for the quietest fans/towers/vents/etc. It is super worth it to get good fans at a decent price with little-to-no sound emitting from the moving parts. Try to get 120 fans, which are bigger and produce more air movement while not being as weak as the smaller fans.

RAM: Get a minimum of 4gb, but 6-12gb isn’t that expensive anymore. Once again, wait for sales.

Power Supply: make sure you get a PS that is compatible with your motherboard. I personally want 500+ watts (anything below may not be able to properly power your computer).

Hard-drives: Since you are on a budget, don’t even think about getting Solid State Drives. You are more likely going to get better results (with your dollars) from getting a 500gb-1tb internal hdd with 7200rpm. Then if you are in need of more space, just get an external hdd of 1-2tb which are actually getting decently priced of 100-200 dollars, but definitely hold off on that extra space until later.

Removable disc media: I personally have a blu-ray writer for around $100, but a DVD-RW is just fine if you are just gaming. Usually below $100 for that, but it all depends on brand and stuff.

Graphics card: I know you didn’t mention this, but like the processor, don’t skimp on this component either. This will probably be your focus since you want a gaming computer. 

Hope this helped. I know I didn’t give specific examples like brands and such, but what I said will help/guide you a lot better than just google/searching “best budget gaming computer system.”