Need help upgrading low tier PC for gaming

Hi. need some help here plz.

so a while ago (year and half) I bought my first PC (that wasn't bought on a wallmart). I wasn't very knowlegeable on the topic and my budget was limited but with some help from the vendor (who is an enthusiast) I managed to get something I was happy with, and I'm still happy, I can play nice games and do stuff, but I think it's time for some upgrading. here are the specs of my rig:

Motherboard: ASUS A-88XM A

Memory: KHX1600C10D3B1/8G x 1
Graphics: AMD RADEON R7 240 2GB DDR3
Storage: SEAGATE BARRACUDA 7200.12 1TB... REF :-(
power source: CORSAIR CX600
Monitor: Samsung 1366 x 768 led monitor, but I mostly use my 40" led TV for gaming (1980 x 1080
Other Peripherals: LOGITECH G600 MMO, KEYBOARD? GAMEPAD!!!
Operating System: WIN 7
Cooling: 3x120mm 1200 rpm Thermaltake exhaust fans (2 on top and 1 upper back), 2x120 mm 1200 rmp intake fans (thermaltake on the bottom and a cooler master in front, with the HDD)

So like I said, I wish to upgrade my rig now that I have the economical possibility, mostly for gaming. I can play most games at low specs, however many current generation games have been impossible to play now, specially since I mostly use my 40" tv for it.

The vendor assured me that the MOBO, CPU and power source were good enough so that I wouldn't have any bottleneck issues if I later decided to upgrade the graphics, however I would like your opinion on what can I expect to put on that rig without having those issues (basically what are the highest graphic card specs I can expect it to run without problems) so I dont waste money on nice card that my rig can´t take advantage of. I do have a personal preference for AMD solutions, so keep that in mind (since intel stuff costs are way, way too inflated on my country, Mexico).

Also what other things would I benefit from upgrading? what would be the most important things to upgrade to step up the game or should I forget about this entirely?

Approximate Purchase Date: Within the month

Budget Range: About 600 dollars (could be a little more)

System Usage from Most to Least Important: Gaming on 40" tv, HD video streaming and reproducing, video rendering and encoding, internet, general use

Parts to Upgrade: Video card and whatever you think its important

Do you need to buy OS: NO

Preferred Website(s) for Parts: Cant really ship stuff to my country so I mostly buy in physical stores

Location: Yucatan, Mexico

Parts Preferences: I kinda like AMD and I think its better for compatibility (AMD CPU, AMD graphics), I can change if you think its best :)

Overclocking: Maybe?

SLI or Crossfire: Dont think so

Your Monitor Resolution: 1920 x 1080 I think

Additional Comments: I'm not very knowledgeable, but I would mostly like to know what look for, so when the vendor shows me their catalog I know what to choose

And Most Importantly, Why Are You Upgrading: I want to be able to play the newer games with decent graphic settings on my TV

Thanks in advance for your time and help :)