Flying in and out of Shannon. 8 days. Have hotels 3 nights on outskirts of Galway, 3 nights on outskirts of Belfast, and then last night near Shannon. Renting car for week. Have already visited the Burren, Dublin, Connemara on a shorter visit last year. Wish I had different arrangements now but this is where we are. Where you go (especially on way to and from Belfast? We don’t mind moving quickly to fit in a lot of sights.
Definitely the Antrim Coast as you are near Belfast. Look for Ballintoy Harbor on the coast road near Giant’s Causeway. This small, picturesque fishing harbour is a short detour from Ballintoy village. Follow the narrow winding road downhill past the white-washed Ballintoy Parish Church one of Ireland’s most photographed churches. The harbour’s stunning natural setting led to it being used as a filming location in the fantasy TV series Game of Thrones®, as exterior Pyke shots and as the Iron Islands. And also nearby is Dunluce Castle. Ruins ,but with stunning views. Both places have small and free car parks. Giant’s Causeway is spectacular but crowded. Best to do this one first in early morning before the tour buses arrive. The other two are not crowded at all as most tourists don’t know they exist. The GoT tour buses can’t go down the narrow road to Ballintoy Harbor.
We were there in July. One outstanding site to visit from Galway would be Kylemore Abbey. And if you get to visit the Aran Islands and the Cliffs of Moher, consider going to the smallest island, Inisheer. It is quiet and so friendly. You won’t find lots of shops, etc. But you can get a cart and pony ride around the island or eat at the local pub. Depends on what you prefer.
Near Belfast you might want to visit Mount Stewart, former estate of the Marquess of Londonderry and now a National Trust property. There's a lovely biography titled Society's Queen: The Life of Edith, Marchioness of Londonderry. She designed the gardens. Her youngest daughter, born in 1921, lived there until her death in 2010. https://www.telegraph.co.uk/news/obituaries/6983261/Lady-Mairi-Bury.html