Belfast girlie here
Belfast is a great city to explore, its small enough that you can walk around it and there is plenty to do and see.
Cathedral Quarter is great for the craic and drinks, usually some live music on and bars all pretty much beside each other.
Some of my faves around there are Duke of York (really traditional), The National, Harp Bar & 39 Gordon street.
There is also a big food market over there called Common Market. A couple of other markets are Trade Market and St Georges Market (only open Fridays - Sunday I believe) but they do a Belfast Bap which will put you into a food coma lol. So good though!
I would recommend going to the top of the Grand Central hotel to the observatory for some views over Belfast in all directions.
The tour buses are a bit of a rip off at £25pp but they are a good way to take in the city and hop off and back on again.
The black taxi tours are decent too for a bit of a more personal tour.
Food wise - Deanes Meat locker, Top Blade, Zen, Dumpling Library, Millars, Fish City are all very good, I really like Villa Italia for cheap and cheerful pasta. Morning Star is good for pub grub, Kellys is good for their homemade stew and Fountain Lane do an unreal carvery on a Sunday
I have heard mixed reports about the Titanic museum in the past couple of years, I went many years ago and wasn't mad about it but I'm sure they have added to it and improved it since I went.
Some other little bars I like are the Deer's Head, they have their own small brewery on site, Kellys Cellars - traditional!, Hell Cat Maggies for views over city hall
I would also definitely recommend the North Coast, you can do the coastal route from Belfast and it is worth it, there is a view point up along there called Magheracross viewing point and it is stunning. Gorgeous wee part of the world with plenty of places to pull the car in, enjoy a wee pint and some local food and take in the views. There really is quite a few places you could stop, I've always liked Bushmills - small wee village with bloody good guinness and you can take in the bushmills tour while there if that is your thing,
Portrush/Portstewart are very popular and can get quite busy but I love it there -
Glenariff is such a lovely place to stop off in,
But really the whole route is signposted so you can stay on the coastal route and you can't go wrong
PS, for Dublin, there is a place called Elephant & Castle and the chicken wings.... I would die for them. So good.