Paul,
If you're planning on going to the west coast of Ireland, then you should fly home out of Shannon. That way, you don't have to double back to Dublin at the end.
For nature, the best place to go is probably Killarney National Park. This is in County Kerry in southwestern Ireland. Gorgeous scenery. The Ring of Kerry and Dingle Peninsula also have spectacular scenery, though not as much secluded, open space as the park.
For history, of course, there's no end of sites: Dublin (Kilmainhaim Gaol, Book of Kells at Trinity College), Rock of Cashel, Cobh (major port for emigration to US).
For shopping, Dublin is your best bet. Lots of shopping in Killarney, but the town is very crowded and touristy.