So if I have a N3DS XL, a WiiU/Wii, a PS3, a PS4 Pro and a Switch currently hooked up in my media center... What's the best way to play S3&K?
I also have a PC that was a capable gaming PC a few years ago, but I don't really use it anymore because it acts funny.
Get Sonic Mega Collection for GameCube and play that on your Wii/Wii U. It has great versions of all the original 16-bit games, including 3&K. Although actually, the Wii VC version of 3 and Knuckles might be a good alternative since the Classic Controller's D-Pad is better than GC's. I've never had a problem using the stick but the D-Pad might end up being more precise or more comfortable. They're both basically the same as far as emulation quality goes.
Then grab the PSN version of Sonic CD and you're golden.