I did a trip with some mates from London when the Euro and the Celtic tiger were roaring so golf at that stage was outrageously expensive in Aussie dollar terms. Given events since then I presume things are more resonanble. My caddie at Lahinch had lived in Australia, we went out for some drinks with them in town and they were taking the piss because I paid as much for one round as they did for a years fees.
I think it's about planning where you really want to play and then working around that as a plan. Royal County Down is a must play, so you know you will be driving north from Dublin, do you push on and play Portrush, perhaps County Louth, that sort of thing.
The drivng is easy for me however I suspect Americans will have a coronary at the "gas" prices. Make sure you car isnt a manual. Given you are on a budget, I presume an escorted tour thing is off the menu.
I played predominately in the southwest, Waterville, Tralee, Lahinch, Ballybunion but as the man with the blog points out these will likely be expensive. Heaven for golfers. You can still fly from Dublin into Shannon for all these.
Good luck, it's a fantastic place. Food is shite, stick to the black stuff.