A lively city. The newly restored Royal Palace is well worth a visit. All the standard tourist things are described in the guidebooks. If you have a chance take a short day trip to Haarlam. It's a pleasant commuter town with some nice squares and architecture, and it's less frenetic than Amsterdam.
Dublin is a short cab ride from where the ships dock and has a bit of something for everyone. St. Stephans Green on a sunny day is hard to beat for flowers, greenery, and just watching the locals enjoying themselves. If you want to see the Book of Kells get their early to avoid the lines. Personally, I think Dublin Castle and the Chester Beatty Library (free)are a better use of time. A lot of history around the Irish uprising is there for the viewing and gives you a good perspective on their more recent history.
The Antrim Coast is absolutely beautiful. The Giant's Causeway is interesting but I don't think worth the drive if you are going direct. The coast drive is what makes it all worthwhile. We used BelfastAttactions for a private cab tour though the city and up the coast. Put over 200 miles on the cab, and it was well worthwhile. Our driver, Girard, for informative, personable and great about showing us out of the way things that we would have easily missed on our own. A nice touch was his iPhone camera which he used at every stop - instant e-mail pictures.
The Georgian and Victorian architecture on the avenues leading from the train station to the river front are delightful. The riverfront and the bridges spanning the Tyne are well worh a visit. The restored riverfront makes for a pleasant walk.
Use the hop on/hop off for local transportation. Hint: If you are doing the Royal Mile. Start at Edinburgh Castle. From there it's all down hill. So much to see and do; manage your time. My top rated site: Hollyrood House, see how royalty lives. They also have a good lunch room. Right across the street is the new Scottish Parliament building. You'll either love it or hate it, but either way go see it and visit the public galleries.