It largely depends on what you like to find. A good place to start is the lists of the most favourited caches by type, and we've got challenge caches that tell you (and then if you really push it and get them all you can you can log the challenge too) GC4RXC2 will give you the current Unknown for Dublin, GC4M87N is the Multi list, and the overall top 20 is here. Ireland's Top 10 is here. You can treat these as an easy way of finding what other people have enjoyed finding. Have a look and see what takes your fancy!
Recommendations also depend on whether all of your party are cachers, and whether you're all equally fit. We've got plenty of 1/1 park and grabs, but we've also got a fair amount of perilous stuff involving boats and climbing gear.
I don't know what you mean by "renting a personal WIFI", but we have pretty good mobile network coverage. Owning a mobile works a bit differently here to the US - we don't need to lock into a contract. Pay As You Go is a prepay system where you get a month's service at a time, no strings. If you have a smartphone, bring it with you and go into any mobile phone store. They'll set you up with a SIM card - a number, basically, and a call and data plan. My current plan costs about Eu20 a month for calls and 3gb of data, but there's plenty of choice. Then at the end of the holiday you remove the card and forget about it. No bills, no rental contract, nothing. And you have a bonus local phone number for any calls you need to make here.
You mention Wales and England - as of now, they're still in the EU, and the EU is due to abolish mobile roaming charges in July. It's possible that you might be able to get a Pay As You Go SIM in the first country you visit and then use it for the whole of your trip in all of the countries at no extra charge, but I'm not sure.
If you do need a separate UK SIM, then getting one is as simple and cheap as here.
If you're planning to drive into Northern Ireland (and I recommend you do - there's fantastic caching up there), let your rental company know. There may be an additional charge on the rental - this varies from one rental company to another.
Organise caching events during your stay - we love them! And it's a great way to meet the locals and get local advice.
If you're on FB, the Irish geocaching community has a very active group there - check it out!