The best (most visible) sections are in the middle around the Haltwhistle to Housesteads area. Corbridge, whilst it does have the fort is two or three miles south of the Wall route and there's none of it left there anyway, most of the stones having been used as foundations for the 'Military Road' built by General Wade in the 18thC.

The bus, the AD122, yes you can get this from Newcastle, outside the Central Station - come out the main entrance and turn right, it's on the main road opposite the steps down to the Metro. IIRC there's a day ticket which means you can get on and off as many times as you like. Stut's already given one link, but here's one to the national walking trail which has the details as well.

http://www.nationaltrail.co.uk/hadrianswall/

Carlisle is not a good place if you want to look at actual remains as that end of the Wall was mainly built in turf and basically no longer exists.

I've walked the whole route a couple of times and while it's not the Inca trail or base camp on Everest the middle sections can be strenuous & do require reasonable hiking foot- & rain- wear.

Having said all that, do enjoy it, it is, from my biased viewpoint, some of the lovliest country in England.