Right, for a start, from Sheffield to Dewsbury, two routes are valid - via Huddersfield (and the Penistone Line) and via Leeds (NOT Doncaster)
I can't help you with how busy they are as I don't commute on the train, but the via Leeds is usually quickest (make sure you don't end up on a Northern service via Wakefield Kirkgate, as they take an age!) - usually a mix of XC (you could get a HST if you are lucky) and TransPennine Express (almost always a 185) - Northern also serve Leeds to Dewsbury twice an hour too, so there are 4 trains an hour on that stretch, they the TPE usually pretty much follows the Northern stopper.
The hospital is a little way from the station so a bus is your best bet (depending which unit in the hospital, even then, you may have a little bit of a walk, as AFAIK, the buses only serve the one part of the complex. The main one I am aware of is the 268 route (which is pretty frequent IIRC), but there are others. Dewsbury bus station is a fair size (some 18 stands) and it also has a travel centre, so you won't be stuck without information, plus there are live departure boards all round the place.
As for getting from the railway station to the bus station, you can either walk (couple of mins walk) or use the Free Town Bus (FTB). Either way, you need to come out of the main station entrance (which is on the Huddersfield bound platform - if you arrive FROM Huddersfield, you need to cross the bridge).
Straight opposite the main entrance is a bus stop. This is where the FTB leaves from, every 10 mins. The FTB from the railway station will give you a full tour of Dewsbury (including our lovely Sainsburys store!) before it gets to the bus station, but this still only takes a maximum of 10 mins (it doesn't usually get very busy)
If you would prefer to walk, the journey to the bus station is all downhill. Come out of the station as above, but head for the crossing directly opposite. Walk down the road to the right after crossing (the "Dewsbury Reporter" office will be on your right, and the registry office and a church will be on your left). At the bottom of the road is a car park - head down the hill to the right of this and the rest will be self explanatory as you cannot then miss the bus station!
If you want to clarify anything, feel free to PM me (it is my natural stomping ground, afterall!) - I can probably get more bus information for you if needs be, but another family member knows the routes better than I do.