How major is a rear derailleur change?

I wouldn't be surprised if the chain/cassette/chainrings were getting worn. that's pretty common. i've never, ever worn out a rear mech to the point where it won't work, i've got a shed full of old ones. front ones rend to seize more easily but even then you can normally resurrect them...

I second what Dave said.

Have you replaced anything on this bike before now? I'd expect the derailleurs to last a few cassettes and cassettes to last a few chains (if the chains are replaced often enough).

Do you actually have any problems with shifting performance? If you do, then the alignment of the rear derailleur hanger would be something else worth investigating before getting a new derailleur.

My first question would be whether it's actually true that they're coming to the end of their life. I've had a few derailleurs off people who've been told by the bike shop that they're dead, when all they needed was a good clean. Chainsets and cassettes wear down because they're transmitting torque but there's not much going on with a mech that'll break it, front ones especially.

My advice would be to remove both mechs and give them a really good clean: soak them in a bath of degreaser overnight and then go at them with some more degreaser/washing up liquid and an old toothbrush till they're shiny. rinse them well, dry them with a rag and then give them a going over with WD40. If you want you can take the rear mech cage apart and put some grease inside the jockey wheels, the metal covers just lift off with a small screwdriver.

After that lube all the pivot points with something that's light enough to really penetrate (ProGold lube is good) and bung 'em back on, with some fresh cables. They'll feel like new, for a bit

keep lubing the pivots every time you use the bike, for a bit, just to work some oil inside. If you do all that and they're still shonky, maybe it is time for some new ones...

It's a fairly straight forward job.

You'll need a chain splitter, allen keys, a new rear gear cable inner, and a tool to cut it to length.

You'll need to remove the chain and the rear gear cable inner, remove the deraileur. Then thread new gear inner, replace rear mech and chain, attach new inner to mech and adjust mech.

The best detailed instructions for adjustment are on the Park Tools site here -

Rear Deraileur Adjustment