My current bicycle is a 2013 Specialized Allez C2. It has an 8 speed rear cassette. If I swap the current chainring (50/32) to a larger one (52/36 OR 54/39), other than the chain itself, what else would I need to change, if anything?
You need to first check that your front derailleur has enough capacity to handle the new chain rings and can clear it (i.e. you can re-position and adjust the front derailleur to handle these chainrings). You should refer to your front derailleur's data sheet to see if this is possible. In general, you also need to check the rear derailleur's capacity to see if the rear derailleur is long enough to take up the extra possible slack, but in this case, the capacity required will go down for a fixed rear cassette, so you don't need to worry about it. You also need to make sure the chain adequately clears the frame under the new chainring size (which may require changing the bottom bracket or other things, if possible).
Here are some directions on adjusting the front derailleur.