What to see while driving between Londonderry and Galway Ireland

Great question- for Belfast to Derry, it depends if you want to go the most direct route or not. If you want to just go directly, you can stop in the Sperrins which is a scenic area: https://discovernorthernireland.com/things-to-do/outdoors-nature-and-wildlife/sperrins

If you have more time, I would go up along the coast where you have all the classic sites along the Causeway Coast.

On the way down from Derry to Galway I would stop off at Enniskillen/Lough Erne. I'm less familiar with the Republic further south than that so will let others suggest the rest of the journey! Good luck with your planning.

I took the bus from Derry to Donegal and thought the Barnesmore Pass was very scenic. Then you run along the coast there through Bundoran and on to Sligo. South of Sligo is the Carrowmore Megalithic Cemetery. A little further south on the N17 is the Catholic shrine at Knock. You could take a detour at Charlestown (north of Knock) and head out to Westport and south from there to Clifden. You'd be adding quite a bit of time to your journey that way, of course. Further south of Knock, you could detour at Claremorris for Ashford Castle.
Taking the other commenter's suggestion of Loch Erne, you could route through Athlone, which has a castle, a historic pub and the nearby Clonmacnoise Monastic Settlement.

There's also the Ulster-American Folk Park near Omagh - would be a bit of a side trip but it's very well done. Might be of particular interest if your heritage traces to Ulster, or if you're just interested in the outmigrations of the early 18th century. Their web page is quite good if you want to check it out and see if it'd be of interest.
I'd also second the suggestion of a stop at the monastic site of Clonmacnoise - another fascinating place and one that's not as overrun as the site at Glendalough near Dublin. Its scenic location along the Shannon makes it pretty easy to envision Viking longboats coming up the river to sack the place a thousand years ago. Would be easy to spend an hour or two exploring the place on foot before pushing on.
And I agree that Athlone would make a good stop for lunch or a coffee.

This involves driving further west, but check out Mt. Errigal and the Slieve League cliffs, both in County Donegal. The former, a hikable, conical-shaped, 2,500-foot mountain can be seen from miles away as you approach from the east. The latter provides dramatic views of the ocean from 2,000 feet up.

https://www.theirishroadtrip.com/climbing-mount-errigal/

https://www.errigalmountain.com/abouterrigal

https://www.slieveleague.com/

remember it will switch speed limits at the border Northern Ireland like the rest of the UK drives in MPH
Ireland drives in KPH

remember the currency switch as well Northern Ireland pounds Ireland euro