Lets set aside for the moment how you're going to "get around".
Time for a Reality Check: you list at least six locations (including some pretty fuzzy ones, eg "a few days in the south area of Ireland"). You say you have "a two-week trip". Do the math. First you need to determine exactly how many usable, full days you will have to spend on the ground in Europe, not including the day you arrive in Europe nor the day you depart. For most people (first time travelers especially), they fail to account for some critically important facts..
A "two-week trip" for many people means they plan to be away from home for two weeks. When you subtract your travel time, that means you end up with 11 (or at best 12) actual usable days in Europe (you can not and should not count your arrival or departure days no matter what time the flights are scheduled). The first day after arrival is often still pretty jetlagged and foggy, but lets set that aside and assume the best possible scenario, you have 12 days to use.
Now lets assume that your "locations" are all singularities - that is, one specific, compact place, that doesn't require much moving around within that location (for example: Disneyland, or Stonehenge). This is NOT a realistic assumption, but just to illustrate how unrealistic your plan is, lets go with it...
12 days, 6 locations. That sounds like 2 days per location. But that is a self-deception, because you do not magically trapsoirt yourself instantly from one place to another. Your locations are scattered across 3 countries and (at least) 6 different regions. And most of all, you need to come to grips with the fact that every time you pick yourself up and move to another location, you burn at best 1/2 a day, but in many cases a whole day. If you're just taking one train from well-connected one city center to another in the same region, that consumes 1/2 to 3/4 of a day. But you are (hoping to) jump around Europe, across long distances, and many (most) of your proposed locations are NOT easy to connect efficiently - even if you know what you're doing. Even for an experienced, efficient traveler, what you're suggesting is impossible in the time you have.
You need to toss out your plan, and come up with a new plan.
You either need a LOT more time (3-4 weeks) or, you simply need to scale back your plans. With the time you have, I'd suggest you pick ONE of the following (not all):
- Ireland (including Dublin and "south area of Ireland")
- Wales and England (including "Brecon Wales and Bath")
- France (including Normandy and Paris)
If you really want to do all the above, you need 3-4 weeks (3 weeks would be very, very rushed).
About "how to get around" (since that's what you asked): To get around on a trip as you are proposing, you need a mix of trains and rental cars, and lots of walking, depending on where you are: Dublin is best on foot (you do not want a car in Dublin or any large city in Europe). Depending on what you mean by "the south area of Ireland") a car works best, but you might include a train or bus in the mix to cover long distances. This is a part of the world that takes some time to get around in. Getting from Ireland to England you can go by ferry or fly. Getting around Wales and England, again, on foot/public trans in the big city, maybe trains to cover distance, and a car in rural areas. Take the train to Paris, and walk/use transit there. For Normandy, you need a car. All of this switching and moving around will consume most of the time you have allocated for your trip...
Sorry, I know this is not what you want to hear. But it's the truth.