As a renter, there really isn't much that you can do to prepare your home for winter, but keeping your drains clear is an easy and cheap way to avoid a potentially much bigger problem. While as a renter you probably wouldn't have to pay for the repairs, you would certainly have to live with the problem, and you just know that the pipes would burst on Christmas Eve or a bank holiday weekend.
Cleaning drains is incredibly easy. You need those two reliable, trusty friends - baking soda, and white vinegar.
First, pull out any debris you can. For the kitchen sink, this may be any food residue around or in the plughole. For the bath or shower sink, I have heard that you can twist a wire coathanger into a hook and pull up any hair that has accumulated in the pipe, but I've never tried this for fear of scratching the enamel bathtub.
Secondly, tip loads of baking soda down the drain. How much you use depends on how clogged your pipes are - do you pour cooking fat down the drain, or have long hair that is prone to shedding? Then pour a bit more baking soda in. Leave the baking soda. While you're waiting, fill your kettle to the max and set it to boil.
Next, pour white vinegar down the drain. Pour it slowly, keeping the fizz up to the top of the pipe as long as you can. Follow the white vinegar with boiling water.
If the drain is seriously blocked, then use a plunger over the plughole after adding the vinegar but BEFORE adding the boiling water! This clip from How Clean Is Your House shows how. Skip to 2 minutes 50.