So, I may want to swap out my current GPU for a better one in the near future. Unfortunately, I know fuck all about hardware, software and computers in general, so I'm gonna need help with that. Here goes.

GPU: AMD Radeon HD 7770
MOBO: ASUS M5A78L-M LX
CPU: AMD FX-6300 @3.5 GHz (+ Scythe Samurai ZZ cooler)
RAM: 2x 4GB DDR3 RAM
HDD: 500GB WD5000AAKS-60WWPA0 (o.0)
PSU: 400 Watt be quiet! Power Non-Modular 80+ Bronze

No additional coolers in case, though I suppose I could, if possible and necessary.

What sort of GPU could I feasibly upgrade to without having to get a new PSU, MoBo or some-such nonsense, which I cannot afford?

I'm not expecting anything that would enable me to play current gen games at high setting; however, if I could play something like XCOM 2 or Dark Souls 3 (shoo, nightcraw1er) at medium or low without anything bursting into flames, that would be a plus.

Let's not talk about budget until I get some suggestions that would go with my current set-up. Like I said, I don't want to replace any other parts for the moment (unless absolutely necessary for a reasonable upgrade, in which case I shall stop spending money on food for a week or so).

First of all you're going to need a new PSU I believe..
Secondly, with new nVidia chips prices of older cards should drop shortly.

What you really need to know is what kind of slot it uses. According to here you have 1 PCIe x16 slot. If it's version 2 or 3 or what i don't know, so that brings down some of the options...

Here's a good place to start. Not sure how to tell it low power consumption since you need 500+ for the PSU assistance. Only if you want it to run with direct power from a PSU, there are mobile chips and decent cards that don't require the extra power supply support, but that's to be seen. If he does upgrade his PSU he should get something like a 600 or 800 watt power supply.

Well, balls. When I upgraded this ragtag rig, the 400W PSU was recommended to me. (Which also means it isn't all that old.)
Do nVidia GPUs and AMD CPUs work well together? Plus, I've always heard nVidia is more compatible with games in general, but AMD is cheaper and about as good with tweaking in some cases. True/false? Many choices, much ignorance on my part. :<

400W might still be enough as long as you go with a lower-draw graphics card. Something like the top-tier AMD cards would of course exceed 400W themselves, but decide first on what card you want, then see if you need a new PSU.

nVidia works fine with AMD CPUs. Very rarely will you encounter trouble from the mixing of brands like that. You're right that in general nVidia has better optimization from games. That's less a technological thing than a human thing, though. You'll be fine either way, honestly. I use nVidia now in my gaming rig, but I've used AMD several times in the past. They're both just fine.

What's your budget? Start there, and we can help you understand what sort of card you might want to consider
EDIT: I wasn't clear. I know you said "let's not talk about budget" but that's like asking an accountant to do your taxes without discussing money. It's not a reasonable request.

SECOND EDIT: OK, I should try to play by the rules. Here are three suggestions and the pros and cons of each:
nVidia GTX 750Ti : slightly faster, lower power draw. Cost you about $120. Not much of an improvement, though, to be honest.
nVidia GTX 960: Noticeably faster. Still can use your existing power supply. Cost you about $200. An improvement, but with Nvidia releasing their new 10xx series, if you can wait a little while you'll see the prices on the 900s drop.
Radeon r9 270: Much faster. You'll need a new power supply, though. Cost you about $300.

I think something like gtx750, perhaps gtx 950 should have about the same power consumption as your current card. But I'd recommend better PSU anyway.

better wait a second and look at those new 1070 things..

Confused face. Goody. Hm, yes, I suppose. Technically, I have about 300€ sitting in a box in case I need to leave the country. Let's say 200€ is the max amount I'd pay for a GPU if I really have to. o_° Is that a literal second or what, 'cause I'm actually impatient as hell, plus, I have no idea what you're talking about and for all I know, it's some sort of NASA technology that needs better surrounding hardware than what I have.

Being old and being quality are 2 different things. The PSU is just low quality. It's like putting cheap oil in your nice car.

You don't have to worry about compatibility like that. Your biggest concern is budget and what you'd like to do. Bursting into flames has everything to do with your cooling system and not with the hardware itself (except a cheap PSU may blow). So if you want low temperatures at 100% performance then you get a decent cooler. If you want better loading times you get a better storage drive. If you want more frames (or for some games any frames) it depends on if its mostly graphics or processor intensive.

You learn through experience so just tell us exactly what you need. Budget's important because you'll be paying $150-$300 for a nice card and another $60-$100-ish for a good PSU. Plus you may need a RAM upgrade and an SSD or SSHD. So yeah, you're gonna go broke depending on your needs. :P

It will not take a much.. a month or so, also 1070 wil be like cost of 970 and will be a lot better. like compared to 980 ti what I've read.

I have though a 500w psu but if my tiny 760 runs a lot off stuff and what I read it takes a bit less power I'mm good (I mean I had money) if you have 400 psu.. I'm not so sure perhaps it is just what quality and what else you have on your machine.

Are you trying to be funny. I'm always broke.

On a serious note, is this PSU really low quality? I thought it wasn't. You live and learn, I guess.

You might as well be speaking Chinese, I honestly do not understand you, based on my knowledge of hardware, which is about 0.
"A month" I get, that seems ... not that bad, actually.

I say wait for a few weeks more, prices will drop : )

The <span class="bold">Asus Strix GeForce GTX 960 OC</span> would be a good candidate.  Its TDP is slightly higher than your current GPU, but nVidia recommends a <span class="bold">400W power supply</span> for a GTX 960 and a Core i7 3.2 GHz.

Power supplies are, for some reason, a slightly contentious issue. It used to be, way back when, that you wanted your power supply to run at about 50% of max load, or even less, most of the time. Parts went bad quickly and heat was a big problem if you kept your PSU anywhere near peak. But power supplies have become a lot more efficient in the last 20 years. What you want to do is look and make sure that your power supply has a good efficiency rating. There's a standard called "80 Plus" which is given to hardware that is at least 80% efficient. If you can get an "80 Plus Gold" rating, that's going to be roughly 90% efficient, and that means you're going to get a lot less waste heat from your PSU, and you'll be able to get an effective power close to the rated power.

I know, that was boring. It's important, though, because it means not all 400W PSUs are created equal. Am anfang, dort war elektrizitat, or something.

They are just releasing those new cards.. Even if you had to upgrade psu a bit I still would buy 1070 than 970. If you are bit patient more I think you can be proud of yourself.