Northern Ireland is part of the UK (United Kingdom of Great Britain and Northern Ireland, to be exact :-) ). They use pounds sterling as their currency. You'll be able to get around Belfast no problem using the buses http://www.translink.co.uk/ You might want to consider the "Belfast Visitor Pass".
You don't mention when you plan to visit. Though I'm from Belfast (and lived most my life in Canada), I have used those "Pay and drive" tours, which includes rental of a car and accommodation at certain hotels/B & Bs. You might like an Ulsterbus tour, depending when you are visiting, if you are not comfortable with the idea of driving on the left side of the road.
First you need to decide:
1) how long am I going for?
2) when am I going?
3) what sort of places do I want to stay at? (hotels. B&B's, self-catering?)
4) how much of the countryside do I want to see (everybody needs to take a trip "up the coast" through the Glens http://www.northantrim.com/theglensofantrim.htmand visit Bushmills http://www.bushmills.com/ ), see Dunluce Castle and Carrick-a-rede rope bridge http://www.nationaltrust.org.uk/carrick-a-rede/ (I don't think I've ever been brave enough to walk over it!)? And if you travel up the Glens, you may as well visit Coleraine and Derry http://www.discovernorthernireland.com/walledcity/ & http://www.derryvisitor.com/
If/when you travel beyond Belfast, where and what do I want to visit and how will I get there?
You can travel by bus to Carrickfergus (locals think the castle a bit ...well ordinary, but it's an extremely well-preserved Norman castle http://www.discovernorthernireland.com/Carrickfergus-Castle-Carrickfergus-P2814 )
Then there is the Ards Peninsula in County Down http://www.ards-council.gov.uk/visitor-information/index.php and Downpatrick (also in County Down) http://www.visitdownpatrick.com/ .
Then there is the Ecclesiastical Centre in Armagh http://www.armagh.co.uk/
There is far more to see and do than can be seen and done in two or three weeks. You need to narrow down your interests. It will be more expensive during the summer. There are fewer flights to Belfast (Aldergrove) Airport this time of year til the spring (except around Christmas and New Years).
You'll have a wonderful time--the people are friendly and very hospitable and the scenery is beautiful and the cities are as safe as any.