Just as a side note, your apartment manager/owner should, if asked, fix the major air leaks with caulking or something, but if you don't want them sealed forever, make sure they don't seal them shut.

Sealing the obvious leaks and applying the different types of weatherproofing so that the window seals off drafts when closed is the first line of defense.

Next, as has already been mentioned here is using the plastic sheeting over the whole window face. Some kits come with a type of plastic that you use a hair dryer on so that you shrink it down to a taut optically clear sheet so you can see through it perfectly. This however locks off the window from being opened at all. Be sure to keep some strategically located windows free enough to be opened for periodically airing your place out. To be warm and live in a poor quality air environment is not a great trade off. Have some way to get cross ventilation or at least one window at opposite ends of your place so you can exchange the air regularly.

Lastly window dressing applications can help insulate your place especially when you are not there or at night (when interfering with incoming light does not matter that much - except for plants and pets). Look at the Warm Window technology that uses nice fabric layers that can be specialized for decor and in the fabric sandwich is an insulating layer that includes a space blanket kind of mylar barrier.

After you've followed everyones' window advice, make sure your window frames and doors are weatherstripped. Get down on the floor to see if you're getting drafts from your outside door, and resolve that if you can (a long snakelike pillow filled with dried beans is useful). And keep doors closed in your apartment so you're not paying to heat space you're not regularly using.