As long as you get a SSD drive that is compatible with your original machine, it's that easy! No other parts need be swapped.
Apple changed hard drive cables with the second revision (A1304) with connectors that are incompatible with the original version (A1237), so you must be sure the SSD drive you buy has the correct connector. The part number is 661-4581 for the original Apple part. Plugging this number in may or may not help.
Good luck!