@xaeniac is your current computer an OEM system (Dell, HP, etc.) or is it something that you have built? I only ask because I had a couple i3-2100 OEM machines that would not allow me to upgrade to Sandy Bridge or Ivy Bridge i7 CPUs and no BIOS updates were available to expand compatibility. They should have been drop-in upgrades or at most require a new BIOS, but no. Best I could do was i5-3450, it wouldn't even post with a 3570K. Just something to keep in mind -
ensure motherboard support for the desired CPU before buying.
The stock Intel cooler should perform fine unless you're going to push
heavy overclocks. Otherwise if you have space for it, the 212 coolers have always been my go-to HSF setup.
It's hard to beat the AMD RX560 or 1050Ti on sale/rebate for absolute value, but a used RX 470 or new 570 is probably the most bang for the buck from a performance perspective. I have seen some cheaper 6GB 1060 cards, too.