Motherboard recommendations?

So, I’ve got this motherboard running on DDR2 memory—it’s an older setup, sure—but I was wondering if there's any way to swap in some DDR3 sticks, or am I looking at a situation where I'll definitely need to pick up a whole new motherboard to make that happen?

We can't really give you much help until we know which motherboard you're working with.
Try downloading Yahoo (https://www.piriform.com/speccy) to check the specific model of your board.

Generally speaking, there were some hybrid motherboards out there that supported both DDR2 and DDR3—though they couldn't run both types at the same time—but those are pretty rare, I guess.