I need to back up my ancient Macbook in order to restore on to a new shiny one
What’s the best/cheapest way to do this?

Would guess an external hd and time machine.

No backup required – just an Ethernet cable to connect the two machines and use Migration Assistant. IIRC… 🙂

I’ve always used migration assistant. Don’t even need to connect them together using a cable. Just fired up the new Mac, it immediately detected my old one on my network and asked if I wanted to transfer over to new Mac.

You may like to make a clone of the old MacBook for archiving reasons. (Use carbon copy cloner, worth the cost I’m told by “The Digital Plumber”)
Portable hard disks are quite cheap and useful for such purposes.
To transfer the user data, though, you can indeed likely use apples migration assistant. It works well.

Later on, that external HD can become your Time Machine backup disk.

What has been said. Migration Assistant for just transfer from old to new, use a bit of cable to make it faster. Long-term Time Machine is fantastic: an external HDD is cheap but most NAS drives support Time Machine and it will just quietly backup for you all the time.

*So useful* to have a backup from 12:34 that morning when your disk dies at 13:10, or you realise that important file has been deleted.