Until two weeks ago I lived in Scarborough, about 1/4 mile from the university, on Prince of Wales Terrace on the South Cliff.
From the spa in the South Bay to the Sea Life Centre in the North Bay will take you about an hour to walk. About half an hour to walk from the University to the Grand.
Depending on what you like there are some nice seascapes to be had in both bays, although of course the sunrise (being on the East Coast) is the best time. Moving around the spa is an interesting old building. The South Bay is the more commercial, if you like people photography you'll see plenty of stuff to photograph there, plus if it is dark all the lights will be on the amusement arcades. The harbour is worth a walk around, and if you go around the outer pier there is some nice views back over towards the Spa.
The Marine Drive is home to lots of seabirds, and some peregrine falcons if you like birding. You've also got some good views to the North with the cliffs and headland to the left. If you're feeling energetic the castle is a reasonable walk up, you've got to pay to get in but once you do there are some good views from the Castle headland.
If you walk right along the Marine Drive when you get to the Oasis Cafe, depending on tide there are some good rocks on the beach to make a foreground interest if you're photographing back towards the castle headland.
Right at the other end there is the Sealife Centre which is quite a striking building, two or three white plastic pyramids.
I geotag my photos on flickr, if you have a look at
http://www.flickr.com/photos/markmullen/sets/72157626315538700/ you can see exactly where these Scarborough shots were taken.
Feel free to give me a shout through my site,
www.markmullenphotography.co.uk if you want any more advice on the area. It was home for 9 years so I know it pretty well!
Have a good trip.