Doolin Base

Corcomroe Abbey is near enough to Doolin to catch on your tour of the Burren. So is Leamanagh Castle though I don't think you can go inside any longer (the building is mostly just a shell anyway). If you happen to pass through Gort, you can see the old tower house that Yeats made his home in, and which features in two of his poems (one of them now inscribed on a slab set into the wall).

You can hire bikes in Aran, as you may already know. The bikes on offer are terrible old clunkers but it's still a pretty nice way of getting around the island. The rental place is right in Fearann a' Choirce where the ferry arrives.

If you are going to be in the West, you should definitely visit Conamara. It's one of the most beautiful and distinctive regions of Ireland, as well as the region that has best conserved the native Gaelic culture. As it's the far side of Galway and is worth a lengthy visit (much more so than Dingle in my opinion, and it's also much nearer where you'll be but again, that's just my opinion), it would be somewhere it would be worth staying a night. Cong (although it's not actually in Conamara) is one possible overnight spot (and also has an interesting ruined abbey as well as the scenic shore of Lough Corrib), or Leenane, or better still any of the coastal villages between Clifden and Kilkieran. The road inland from Leenane down past Finney and Loughnafooey towards Maamtrasna and Lough Mask is also very scenic. There is some great cycling in the region, for example I had a great ride from Carraroe out west heading straight across the bog road to Cashel and then back in the loop along the coast through Carna and Kilkieran.

But I'm afraid I don't know where you can hire bikes in Conamara - I brought my own. It wouldn't be worth hiring bikes in Galway itself, unless you are very fast cyclists, because the first 30km or more out of the city is very heavily trafficked and not particularly scenic.

Fear rua has offered some great ideas. I would add that if you enjoy gardens, Dromoland Castle (near Ennis) and Ashford Castle (Cong) have beautiful grounds. Ennis is worth visiting and the hostel there rents bicycles. A couple years ago I was on a bike tour with friends, 2 of whom used those heavy old beater rental bicycles for an entire 2-week tour! So if into biking, it's definitely possible. The Burren is a great place for a day tour on a bike, as is the road out to Kilmacduagh monastery near Gort. A ride down the coast from Doolin would also be good. If visiting the Yeats tower at Thoor Ballylee, Coole Park is also worth seeing. Very near Ennis is Quin Abbey, one of my personal favorites. Dysert O'dea or Dungaire Castle both offer the type of tour where you can go in and see what life was like.

Roundstone would be my pick for a favorite Galway coast village, but the entire coastline drive is beautiful, as is the route through Connemara, taking in Kylemore Abbey and maybe Ballynahinch, a wonderful estate hotel with more beautiful grounds.

I think it would be very easy to fill 6 days without going so far as Dingle.

Note that if you're relying on Google Maps to look up any of the places mentioned above, you have to zoom in almost all the way before the smaller local roads are marked on the map.

try Connemara
Youtube the following : AT YOUR SERVICE PART1
THEN PART 2 AND PART 3
enjoy.

I missed the bit about hiking. There are also opportunities for this in mountainous parts of Galway/Mayo (as well as in Wicklow near Dublin). There should be information on this in the Lonely Planet or whatever guidebook you have, but when you get to Dublin, you should see if you can pick up the book 'Best Irish Walks' by Joss Lynam, which has some great ideas for hikes. Lynam was one of Ireland's best known mountaineers, but the walks in the book are suitable for anyone reaonably fit who knows how to navigate with a map and compass.