If my flight arrives in Bristol, what kind of transportation connections can I get to the university residences in Bath? I'm going to have a lot of luggage since I'll be living there for 9 months.

Take a look at this :
http://tinyurl.com/2hb8vl

That's easy just outside the exit you get the shuttle bus to Bristol Templemeads train station then get a train to bath.
Shuttle bus takes about 25 mins and stops outside the train station.The train takes about another 15 mins. The shuttle bus runs about every 15 mins so don't panic if you miss one. The trains from Bristol to Bath also run very regular

Bristol might be (just) the nearest airport to Bath, but it really isn't the most convenient for most people and has very few flights to anywhere.

Most major cities in North America have nonstop connections to Heathrow or Gatwick: a number of Canadian cities (and a few American ones) have discount nonstops to Gatwick, mostly via Zoom.

There are reasonably frequent and fast direct buses from Heathrow to Bath (www.nationalexpress.com). The handiest way to get from Gatwick to Bath is by train, changing at Reading

The OP may be flying Continental from Newark.

Newark's not the most obvious place to start a journey from Saskatchewan. Nor, if you live near an airport served by Zoom or flyglobespan (as most Canadians do), the most economical.

Thinking about that Zoom also now fly to Cardiff so that could be aother option. Shuttle bus from Cardiff airport to the main train station and then train to Bath. It is just my thoughts but national express bus takes ages to get anywhere.