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.
Luigi's in Campsie has good, cheap food, and a colourful proprieter. Zio's on Main Street is also cheap, and reasonably tasty.
Check out the Kamal Mahal, also in Campsie, between twelve and two in the afternoon.
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.
Sally's or Top of the Town. Basically just go to John Street and take your pick.