Changing money is not an issue, all banks, most travel agents and lots of Post Offices will do it for you. Alternatively you can pay with credit cards and take your money out of ATMs.
The North is also cheaper than the Republic. Food and drink, accommodation, all better value in the North. I second the Giant’s Causeway, and suggest a stop at the Bushmill’s Distillery nearby. If you want a recommendation for places to stay on the North Antrim coast, I can recommend the Causeway Hotel or the Downhill Hostel as good places.
Go to Dingle driving over the Conor Pass. My recommendation to stay is Crutchs Hillville House Hotel. It’s a bit mad, but it’s clean, not too expensive and the owner will cook you whatever you want for breakfast and dinner- oh yes, and he’s called Ronald MacDonald, so that’s a plus.
Connemara is a lovely part of the country- very wild, and not quite as touristy as Kerry. For a bit of a treat try Renvyle House. Currarevagh House is another option, my parents love staying there. Do bear in mind that my family like homely, slightly eccentric places where you can lie on a sofa and order food and drink all afternoon before going sightseeing, taking a bracing walk or playing a round of golf. We’re not big on “boutique” or “designer” hotels.
Galway is a great city, it’s small, fairly bohemian and has a nice atmosphere. Limerick and Cork are skippable.
Dublin is obviously going to be on your list. For somewhere nice to stay, try the Pembroke Townhouse, or for somewhere cheaper, choose one of the many hostels- Avalon House and the Isaacs Hostel are where my friends usually stay when they’re in town.
No time is guaranteed to be sunny, but June-August will be the most likely to give you some good weather. Bring everything from swimsuits to waterproof jackets and umbrellas. Dress in layers, because the weather is very changeable. It could be wet, dry, hot, cold, foggy, clear or any combination of the above in a single day.
It rarely drops below freezing, but you don’t want to be stuck halfway up Errigle, wearing only a t-shirt, when the heavens open and the wind starts up!