Should I only consider AMD graphics cards for my AMD processor?

simple answer: no. which card to get it depends on you. do you want the best bang for buck? or do you want specific feature offered by those gpu

No, you can get any Card over Any Brand CPU
That budget is too much.

I would say use some of it to Upgrade your CPU
A APU will not be able to handle a $400 GPU at all.

So get a better
PSU
GPU
CPU
Mobo

I mostly just want performance. I'm a 1080p gamer who wants to be able to play games like Battlefield 3 with at least High settings. I'm pretty much just getting a new graphics card for gaming, not really for anything else.

Also, what do you think of the aforementioned computer? Do I need an upgrade or will I be fine running a ~$400 graphics card?

A radeon hd 7850 would be fine for $160 bundle that with a corsair cs 430 and you will be set for 1080p high for under $200

^+1 with him, need i5 build

your amd apu will not sufficient enough to deliver good performance

it will definitely going hold back high end gpu such as amd 7900 series. APU are made for those who want simple working dekstop with better multimedia capabilities that provided by intel igp. if they want to do to a little bit of gaming (not the hardcore type) they can add entry level gpu to the system and enable amd dual graphic to boost the gaming performance. that's about it when people choose to use APU. you can add much better gpu to it but at some point the cpu processing power will not be enough to keep with the gpu specially high end cards like amd 7900 series.

if you insist on keeping your current setup i advice you to get mid range level gpu.

Do you think I can edit my existing setup without problems?
I'll need a better CPU and power supply for sure, as I have an A8 Trinity and only 300W. I never knew how clueless I was. I only got this computer in August, when my father got it for me as a surprise. I think I'll at least have to keep this computer as a base.
Is a new motherboard completely necessary as well? Sorry for the lack of knowledge, as I said I only really got into PCs in August.
~$230 of my budget is Best Buy credit, by the way. I was looking at a MSI R7870 HAWK before I got a larger budget.

Thanks for the list, but is the motherboard completely necessary as well? I'm very new at this. And do you think I can put these parts in my existing computer without problems?

The motherboard is NECESSARY, you cannot put a Intel processor on a AMD socket, it doesn't work like that. The parts should fit in your case so it should work. Go for what Sumukh_Bhagat said, i5 will be way more powerful than that awful APU. The GTX 660 should be able to run BF3 at decent settings on 1920x1080.

One more thing. If I could get a MSI R7870 HAWK, should I go for that or stick with the 660?

Both are solid choices, the 660 is limited by a 192 bit bus though. I would just go for whatever is cheaper.

Thank you all so much for the input. I think I'll go with what Sumukh_Bhagat suggested, although I think I'll go for a MSI R7870 HAWK instead because I've heard a lot of good things about it and I've even heard it outperforms the regular 7950. Any suggestions are still taken, as I haven't bought anything yet and am probably waiting a little bit before I buy a graphics card.

Best answer selected by Sicariis.