Hourly buses from Belfast and Derry can take you to Omagh, passing through other towns on their way. For example, the Derry bus stops in Strabane, Sion Mills and Newtownstewart, while the Belfast buses stop in Dungannon and Ballygawley.
Most of Omagh can be explored on foot. There are no trains or trams, and buses connect most of the suburbs to the town centre, but there's nothing to do there. So yeah, walking around is your best bet.
No visit to Omagh or the surrounding areas would be complete without dabbling in the local tradition of streetdrinking. This is usually accompanied by copious amounts of Buckfast, a tonic wine produced in Devon, but insanely popular in Scotland and Northern Ireland, as well as the Republic. Popular drinking places include the Grange Park and the Riverbank. Give it a try!
Luigi's in Campsie has good, cheap food, and a colourful proprieter. Zio's on Main Street (called that even though you wouldn't think it was a street at all) is also cheap, and reasonably tasty.
Check out the Kamal Mahal, also in Campsie, between twelve and two in the afternoon. All you can eat lunches for a fiver, with awesome food.
In Newtownstewart just outside Omagh there is a great local restaurant called Mambojax, it concentrates on local suppliers and has an amazing menu, its a real taste of Northern Ireland.
Lots, because there isn't much to do otherwise. Omagh is a pretty expensive town to drink in. It's much cheaper to get a 'carryout' ( a local word meaning alcohol puchased in a shop), as bars in the town rarely have drinks promotions.
If you decide to venture out, The Embankment and the newly-opened Rue are in fierce competition for the title of "the place to be seen", Other pubs such as Sally O'Brien's or Top of the Town offer the useual guiness and craic that you would expect from a bar in Northern Ireland.