First, check your powers upply's label and see how much wattage it can shell out? For any future upgrades you will need a 500W to be good. That is more or less also the minimum powersupply recommendation for most of todays cards. A modular powersupply will save you a lot of cable clutter. Corsair, are good ones for example. Never step down in quality on a psu and buy some no name stuff for like 40 bucks!
If you do? It will come back and haunt you pretty soon. Used powersupplies are not a good way to go either. You will have no idea of the mileage or the condition.
If you dont have a small form factor case? Then this card is a good choise for a small sum: EVGA GeForce GTX 650 01G-P4-2650-KR Video Card - 1GB GDDR5, PCI-Express 3.0(x16), 1x Dual-link DVI-I, 1x Dual-link DVI-D, 1x Mini-HDMI, DirectX 12, Dual-Slot - 01G-P4-2650-KR at TigerDirect.com
The integrated video chip are often just enough to run everything but games for most of the time. At least the slightly older chips that is. And might automatically be disabled in bios when you stick a dedicated card into the slot. Othervice, disable it yourself. Keep the card you have got now as a backup. It will keep you going if anything bad happends to the card you are going to get in the future.
Make sure to uninstall the old cards driver before you install the new cards software. The old driver may cause problems othervise.
EDIT. You will have to get a new power supply. The one you have got is only 250W. The case are ok. for any modern card as it's a mid tower.