Hi Margaret. When you say "no tours", I assume you mean you don't want to be on a multiple-day escorted coach tour.
While you can get from the large cities and towns fairly easy with bus or train service, to get out to the more remote sightseeing in the countryside, you'll either need your own rental car, hire a private driver, or look into day tours from the base cities or towns along your route. Research how long the estimated travel times are between the cities/towns you want to visit, but then add more time to the estimate. Travel in those countries can be quite slow, even when you leave the driving to someone else.
5 days per country is tight with all the travel connections. You'll get a brief taste of a few select places unless you are willing to pare down the number of countries you want to visit.
June is shoulder season and a nice time to travel to avoid large crowds and have an easier time booking affordable accommodation while still enjoying decent weather.
There are many options for accommodation in Ireland or the UK. It all depends on your preference and budget - large hotel, guesthouse, B&B, hostel. Read reviews on sites like TripAdvisor.com once you narrow down what cities you'll be visiting and the type of accommodation you like best.
For must-see's - start by checking out some guidebooks and travelogue DVD's from your local library. Rick Steves, Fodors, Frommers, etc. If you have access to iTunes, download free podcasts (including ones from Rick Steves) to listen to discussions and recommendations on things to do in various countries. Lots of episodes of Travel Channel shows like Anthony Bourdain's "No Reservations" or Samantha Brown's "Passport to Europe" can be found online to help you plan your trip.
You've got plenty of time to research and come up with a rough itinerary that suits your wants & needs, then ask specific questions about your intended plans.