It is possible to do the whole network in one day but it’ll be a long day. (Belfast to Derry is 2.5 hours each way). If going to Derry sit on the right hand side of the train. The bit from Coleraine to Derry has great views.
Other tourist things:
A drive through the Mountains of Mourne in Co Down
Visit Downpatrick Cathedral to see where St Patrick is buried (allegedly)
Mount Stewart stately home near Newtownards (home of the Marquess of Londonderry) or Castle Ward at Strangford. (Both in Co Down). National Trust (free admission for those with reciprocal arrangements).
http://www.nationaltrust.org.uk/mount-stewart/
Open top Bus tour of Belfast
Crown Bar in Belfast (200 years old, gas lit. Very popular with visitors). Has featured in many films eg Odd Man Out.
http://www.nationaltrust.org.uk/crown-bar/
Traditional Irish singing and fiddly-dee music in Fibber Magee’s bar (next to the Crown Bar). Most nights from 9pm onwards. Free admission.
Guided walks available. Contact the visitor centre in Donegall Square.
http://www.ireland.com/en-gb/amazing-places/northern-ireland?gclid=CIOjr8CKisICFc3HtAod-iIA0Q
If you like good seafood, go to the Mourne Seafood Bar in Belfast. (Booking pretty well essential in the evenings but less so at lunchtime).
http://mourneseafood.com/belfast/
Tour of Crumlin Rd Jail in Belfast, which closed in the late 1990s. (Includes the execution cell, with working gallows, and an opportunity to lie in the empty coffin, if you are so minded).
http://www.crumlinroadgaol.com/public-tour.html
Bushmills Whiskey distillery tour (2 miles from the Giant’s Causeway). Interesting tour and an opportunity to meet lots of American visitors.
Londonderry/Derry is well worth a days visit. Contact the visitor centre for details of guided walks. Walk round the city walls (the only completely intact set of walls in any city in Europe). See the guns used at the siege of Derry. I recommend Martin McCrossan’s tours..
derrycitytours@aol.com See St Columb’s Cathedral. Visit the Bogside and see the murals. Learn about the more recent conflict in the area between Protestants and Catholics. Buy some corny souvenirs in the many tourist shops. The city is very compact and easily covered on foot. Walk over the peace bridge from one side of the river to the other.
Carrick a Rede Rope Bridge Co Antrim (a few miles from the Giants Causeway). Wobbly rope bridge over a 200 foot drop, leading to a small island. Originally built for salmon fisherman. Now a major tourist attraction. (Most visitors do Bushmills distillery, the Giant’s Causeway and the Rope Bridge as part of the same excursion). Check the National Trust website as you sometimes need to book due to excessive numbers.
http://www.bushmills.com/BMTheOldBushmillsFindUs.html
Carrickfergus Castle. Carrickfergus was the original capital of Ulster before Belfast. The castle is 800 years old. It was a military base until very recently and is intact.
http://www.carrickfergus.org/tourism/attractions/carrickfergus-castle/
Ulster – American Folkpark in Co Tyrone.
http://www.nmni.com/uafp/What-s-on