Donegal, Sligo, Leitrim and Mayo, wilder, cheaper and less touristy than other places like the ring of Kerry.
I personally prefer the west coast as it's wilder and more remote but assuming you arrive in Dublin, the Wicklow mountains, Carlingford/Greenore and the Boyne Valley are places not too far away from Dublin which offer a lot of beautiful scenery.
I'll take an opportunity to plug my blog but it might give you some ideas if you want to go north/west - https://theoldbikeshome.wordpress.com/ I'm happy to advise if you want any more info on any of those places.
I personally prefer the west coast as it's wilder and more remote but assuming you arrive in Dublin, the Wicklow mountains, Carlingford/Greenore and the Boyne Valley are places not too far away from Dublin which offer a lot of beautiful scenery.
I'll take an opportunity to plug my blog but it might give you some ideas if you want to go north/west - https://theoldbikeshome.wordpress.com/ I'm happy to advise if you want any more info on any of those places.