Given your timescale I would probably go directly to Galway/Mayo, then work my way up along the west coast. Going much further south would leave you driving a lot with little time for activities.
Achill is nice for cycling around. Be sure to check out the famine village – you’ll see loads of abandoned homesteads and small holdings all over the WAW but this is one of the bigger sites.
Belleek Castle in Ballina is a very strange and entertaining place to stay, on the way. Dodgy rooms, but awesome food, bar and tour of the tunnels beneath. Up towards Sligo, Benbulben and Yates country has some beautiful scenery and really nice hiking / cycling.
Then definitely check out Slieve League, best sea cliffs in Ireland. Local villages have nice pubs too. If you have time and you can take it in check out the Bluestack Mountain Drive – spectacular loop with interesting spots along the way. Definitely check out Glenveagh and climb Errigal if you can, probably some of the best views in Ireland to be had there.
Possibly a trip out to Tory Island. Ballymastoker, Portsalon, Knockalla….loads more.