Looking to upgrade my PC piece by piece overtime! But no idea where to start!

###Build Help/Ready:

Have you read the sidebar and rules? (Please do)

Yes

What is your intended use for this build? The more details the better.

I'm wanting to improve my FPS in games as it has fallen a lot lately. I don't need anything top of the line, but I multi task a lot so having a game open and twitch on my other monitor isn't uncommon. But I'm a bit short on funds so it will be a piece by piece process as I save. So I'm mainly looking for what would be the best thing to upgrade first, and second and so on. I'm not very good with computers so any bonus information is really appreciated.

If gaming, what kind of performance are you looking for? (Screen resolution, FPS, game settings)

My sole focus is FPS. I wouldn't mind increasing my settings but I generally don't notice much of a difference myself and being on a budget I really just want to be able to play the game while watching Netflix or something.

What is your budget (ballpark is okay)?

For right now, and for the quickest upgrade I was hoping something $50-$150. I can save for more if it's needed, but I'm trying to go as cheap as I can. I don't mind watching for sales either on r/buildapcsales. But I just don't know what to watch for and a few different options wouldn't hurt.

In what country are you purchasing your parts?

Ohio, United States.

Post a draft of your potential build here (specific parts please). Consider formatting your parts list. Don't ask to be spoonfed a build (read the rules!).

Well, as I don't really know what I'm looking for, I figure I'll just put here what my current PC is. If you need any information on more of these, please let me know I'm honestly looking all these up to input them as I don't know and I'm not sure if I'm providing all of what is needed.

CPU- AMD Phenom II X6 1055T Processor 2.80 GHz

Motherboard- GA-880GA- UD3H

Installed Ram - 8.00 GB

Video Card- AMD Radeon HD 5670

Provide any additional details you wish below.

Well I guess this is it! Any advice/suggestions/free money (lol JK) are welcome! If you need any more information please let me know! Thanks so much for any help!