The Geforce 610 GT is a low-end graphics card. Replacing it with a better one would give the greatest change. One suggestion: Geforce 750 ti.
Things to consider, to avoid frustration:
Do you need a low profile card? (The 610 is available that way. If you need a low profile card, that will severely limit what you can use.) Your motherboard is full sized, so it's likely that the case doesn't need a low-profile card.
Does the card have the right connectors to work with your monitor? (DVI, VGA, Display Port, HDMI.) That's one of the least likely problems, but that should be checked.
Does your power supply support the card? (Probably needs to be 300W or more. There is a plaque on the power supply that gives its specs. You may be able to read it with the cover off. If not, Grandpa may recall what power supply he used.)
Does the card require additional power connectors? (Some 750 ti cards don't. That simplifies things.)
This is the least expansive 750ti card at Newegg:
ZOTAC GeForce GTX 750 Ti DirectX 11 ZT-70605-10M 2GB 128-Bit GDDR5 PCI Express 3.0 HDCP Ready Plug-in Card Video Card - Newegg.com
($105US plus $3 S&H). It doesn't have a VGA output, but it includes a DVI to VGA adapter, so that wouldn't be a problem.
I hope that I'm not intimidating you. It may be possible to find a used card that would be OK, but all the same considerations would apply. An old 8800 GTX might be adequate, but it's a large card, and it would require a 450W power supply with two 6 pin PCI-E power cables.
I hope that your grandpa doesn't feel that you're ungrateful because you want to turn your PC into a capable gaming system.