Welcome to GFE,
waterfordgirl.
Just two questions: where are you flying from? and What time of year are you travelling?
In terms of start and end points, you could fly into and out of any points you like, depending on flight availability from your destination and your eventual itinerary. As an example, you could fly in to London and out of Dublin, or you could do a round trip to give yourself more time in London as 2 days will not be enough, even though it's not on your must see list.
Dublin for 2-3 days is good. 2 full days would be fine, depending on how much you want to see, so if you have a lot planned, and it is on your must see list, see if you can make it 3 full days and you can take your time and enjoy the city.
Cork - really only needs 1 day, which will include time to see Blarney Castle.
Killarney vs Galway - If you go with Galway, make it for 2 full days as you can take one day to visit the Connemara and/or the Aran islands. Killarney is a pretty little town (great pubs and chop shop!), and is located in a beautiful national park, so would be good if you plan to travel in Summer. Even 1 full day would be a good start, but if you enjoy hiking, mountain biking, etc, then choose 2 days in Killarney instead.
Belfast - 2 nights, or 1 full day, will only allow time to explore the city, and there is much to see. You won't have time to get out to the Giant's Causeway with only 1 day.
Glasgow - 1 full day is good for Glasgow, you can explore the city ( and hop-on hop-off city tour bus is a good start) and take in a show or comedy night in the city.
Edinburgh - 2 full days would be good here. Stirling for 1 full day will also let you explore, but you could also stay in Edinburgh for an extra night and take a day trip to Stirling and Falkirk instead. Unless you have family links in Aberdeen, I'm not sure you'd enjoy a full day here. I note you haven't listed the Highlands in your itinerary - any reason?
Liverpool - 1 full day is plenty of time, again unless you have a full list of things to explore in Liverpool. It's a fun city, so a full day and a night to visit the Cavern Club would be fine.
Birmingham vs London - to me there's only one choice: London. And as mentioned, I'd make it for at least 3 full days which will allow for sightseeing in the city (no day trips), depending on your interests. Birmingham again is fun, but London should not be missed.
Cardiff - great town, castles, stadium, pubs, Eisteddfods, old meets new architecture. 1-2 full days would be fine.
Brecon - a small Welsh town which is a good access point to the Brecon Beacons if you plan to hike. If so, you'll need to assess your hiking route to work out how much time you need here. Depending on the eventual route you take, you could also travel through Conwy on your way to/from Ireland, which has beautiful landscapes and the odd castle.
Travel methods will depend on your preferences and travel routes - bus and train travel around Ireland are possible, but driving yourself (if you feel confident) is a great way to operate your own timetable and get off the beaten track if you spot something you want to take a closer look at. This is particularly relevant to Ireland.
Train travel in the UK is expensive when buying Point-to-Point tickets, so look into a BritRail pass at
www.britrail.com. Alternatively, National Express is the bus system between cities and is significantly cheaper than train travel, but is also slower. You can also look into flying between, say, Glasgow and Belfast or Dublin and even back to London depending on your final route. Ferry travel between Ireland and Scotland or Wales is also possible.
You can book ahead for B&Bs or, depending on the time of year you are travelling, you can simply book at the train station booths (located in most places upon arrival) for budget accommodation. There are also B&B Guides available in good bookstores for Ireland, Scotland, Wales and England which will allow you to plan ahead. Online services are also an option, even just to get an idea of costs, locations, etc. Hostels are available in each location you've listed, again just do an online search for options, but keep in mind that the busy period (May-Sept) means you'll have to book ahead and early if you're travelling at that time.
Once you've had a chance to set your actual order of itinerary, let us know and we'll be able to suggest transport options for each sector.