Here is a list of supported processors for your motherboard:
http://www.ecs.com.tw/ECSWebSite/Product/Product_Support.aspx?DetailID=1478&CategoryID=1&DetailName=Feature&MenuID=24&LanID=0
It looks like the motherboard is limited by the wattage 0f the processor.
Your G2020 has a passmark rating of 3414 and a single thread rating of 1132.
For most games, the single thread rating is most important.
Modern games will run better with 4 threads vs the two you currently have.
One option is something like a I3-2120 3877/1685.
At the other end would be a I7-3770S 8889/2021.
Google other options with the keyword passmark to find their ratings.
If your games are multiplayer, the total passmark rating is more important.
On the graphics side, fast action games need a very good graphics card.
I would start with at least a GTX1050 card.
How strong of a card you can use will be limited by your psu.
Here is a chart for your options:
http://www.realhardtechx.com/index_archivos/Page362.htm