How do I run a Bitcoin node?

A: by downloading "Bitcoin Core" on your computer.

Installing it appropriately, & running it every so often to keep it current - if you keep it running 24/7 you're doing the network a bigger favor, but it's not a must.

if you have 2TB to spare i'd run a full node, if not just prune it to 5-10GB - it gives you that option at setup just don't click "next" frantically like a baboon 🤣.

Should I do it on my old laptop?

A: Yes.

Should I also mine?

A: this has nothing to do with that, but definitely.