I have a Shimano compact double on my road bike with an 11-28 Cassette (50-34 Chainrings). I would like to change the cassette to 11-32. Is that possible and if it is are there other components I need to change?
You'll need a new chain, since the chain will be slightly longer.
You need to see what the ratings of your rear derailleur are. It has to have enough capacity (i.e. the capacity rating has to be greater than or equal to large chainring - small chainring + large cog - small cog) for your setup. It also has to have appropriate ratings for the largest and smallest cog (i.e. the smallest cog rating has to be <= your smallest cog on the cassette [this is practically never a problem], largest cog rating has to be >= the largest cog on your cassette). If your RD meets these, great.
If your rear derailleur doesn't meet these, well, you need a new one. Here you have to match the cable pulls. Shimano 7-9 speed have the same cable pulls between road and mountain so you can use any RD in the 7-9 speed case, if you have 7-9 speed shifters. Shimano 10 speed mountain uses a different cable pull than Shimano 10 speed road (which is the same cable pull as 9 speed).
My tiagra on giant defy 2014 handled the change from 30T to 32T maximum without too much fettling needed.
It depends of derailleur and bike angle where they meet too, you need to adjust B screw so that the top wheel of the derailleur is moved 'back' from the cog
It's one of the most popular questions on bike forums, and i looked at a lot.
In the end you have to decide if you can do it yourself and buy a new cassette (and then new longer chain) cost around $50? for both, or just ask your bike shop, let them take the worry/hassle and pay for their labour.