Where to start....
1. I think I would first start with the graphics card.
As a test, Run YOUR games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.
If you are budgeting about $150 for a card, a GTX1050ti will run off of slot power and not need a psu replacement.
What is the make/model of your psu?
2. 4gb of ram is simply not enough.
8gb will be minimum.
If your current kit is 2 x 2gb, you will need to replace it with 2 x 4gb or a 2 x 8gb kit.
If you have a single 4gb stick, you could take a chance by adding a similar additional 4gb stick.
Such compatibility os not 100% guaranteed, more like 90%.
3. When it comes time to upgrade the processor, you have some choices.
Your I3-3320(you mean i3-3220) has 4 threads and a passmark rating of 4230 and a single thread rating of 1761.
It is the single thread rating that is most important for most games.
The i3-3220 is a surprisingly good gamer.
Here is a list of processors that your H61 motherboard can support. https://www.gigabyte.com/Motherboard/GA-H61M-S-rev-10#support-cpu
The strongest will be a I7-7700K.
It is an 8 thread processor with a passmark rating of 12096 and a single thread rating of 2581.
You will probably need to pay $250 for a used one on ebay.
If you are inclined to upgrade to a more modern platform, then your options are ryzen or intel kaby or coffee lake.
Ryzen was popular because you got many threads for a decent price.
But, for gaming, 2-3 threads is sufficient for most games and the superior single thread performance of Intel is better for most games.
A $85 G4600 has a passmark rating of 5141 and a single thread rating of 2052.
$190 buys you a 4 thread i3-8350K with a rating of 9641/2408.
On the ryzen side,
$130 buys you a 4 thread 1300x with a rating of 7328/1860
$200 buys you a 12 thread ryzen 1600 with a rating of 12335 and a single thread rating of 1831.
In any of those cases, you will need a new motherboard in the $70-$150 range
and DDR4 ram at $75 for 8gb.