I want to spend 2 months in Ireland next year (maybe August and September or September and October.) I won’t have a car. I’m looking for a base to venture out on daily walks and I’d like to be in a town or city in which I can walk to stores, pubs, etc. But also, since I’ll be there a while I’d like to be able to travel to other parts of Ireland or even catch a cheap flight to the UK. Any suggestions?
As a start this is a list of the airports in the Republic of Ireland-
Dublin (obviously, for almost anywhere),
Cork- for Bristol, Glasgow, London–Heathrow, Birmingham, Edinburgh, Liverpool, London–Gatwick, London–Luton, London–Stansted and Manchester,
Shannon- for Birmingham, Edinburgh, Liverpool, London–Gatwick, London–Stansted and London-Heathrow,
Knock/Ireland West (in County Mayo) for the eponymous shrine- flights to Birmingham, East Midlands, Edinburgh, Liverpool, London–Luton, London–Stansted, Málaga, Manchester and Bristol,
Kerry - for London Luton and Stansted, also Manchester,
Donegal - for Glasgow (on Loganair). Being Loganair these are NOT cheap flights. Bus to Dublin Airport then fight would be way cheaper. Nonetheless the option is there.
We were in Ireland and Northern Ireland in September/October 2018, but had a rental car. Have a good raincoat, and maybe an umbrella, too.
Without a car, I’d suggest staying in Ennis, in County Clare. We stayed there for a big music festival last year (in the Spring), enjoyed walks, and it’s an easy, quick bus ride to the Shannon airport for connections to the UK.
One way to approach this is to take a look at daft.ie and airbnb to see what accommodation is available that suits your needs (in-town location, size, cost, proximity to where you want to walk--the beach, the "mountains" or the countryside?), and then take a look at whether the town/city has the local and long-distance transportation links you need. You can see the train map here: https://www.seat61.com/reference/Ireland-map.pdf?dl=0
You'll see that the routes fan out from Dublin.
Otherwise, transportation is by coach, and there are several operators.
Keep in mind the ferries, as well. They go to Spain, France and the UK.