9 Days should be plenty of time.
I would try to fly into Shannon and out of Belfast is possible(or Belfast-Shannon and reverse the trip)
Day 1 Galway
Day 2 Connemara staying in either Clifden or Westport (about a 3 hour drive and so beautiful you won't mind it at all)
Day 3 Sligo Town (2.5 hours) I've never stayed there but have driven through it a lot.
Day 4 & Day 5 Donegal- Oh...Dear Donegal. It's a big place and a lot of it is rural farmland (like co Kerry) so you have to pick where you want to go.
Killybegs, Gweedore and the Innishowen Peninsula are my favorite places. I highly suggest taking a day to drive the Innishowen if the weather is nice.
Day 6 Derry City- (1 hour from Letterkenny Co Donegal)
Day 7-8 Antrim Coast (about 7 hours to drive the entire coast Derry-Belfast but I would stop along the way in a coastal town for the night)
Day 9 Belfast City