You are right to say that the compact crankset will give you smaller gears to help you get over hills.
You can do a straight swap of the crankset, everything else will be compatible, although will need readjustment.
If you currently have a 5600 crankset, then you are absolutely safe to swap for another 5600 (compact) crankset. Now, the compatibility between different Shimano ranges is governed by the number of sprockets on the cassette, and the good news for you is that both 5600 and 5700 are 10-speed. So, basically, 5600 or 5700 cranksets will fit you perfectly (as would any other Shimano 10-speed road crankset).
5800 is an 11-speed system, you could probably get away with using this crankset (since the differences are small), but it will not fit as perfectly as 5600/5700.
So, looking at Shimanio's site, the part you need is either the FC-5750, or the FC-5650. The "50" appears to denote compact.
Lastly, do you know what length your cranks are? It might be worth measuring this just so that you get the same length with the new crankset. (You'll see on the Shimano site that things are available in four different lengths, the differences between each are small but they will affect your fit.) Here's an example of what you're measuring: