Hi
Both words (verbs) are right spelled and means something like "to be glad or happy". Alegrarse is much better than Regocijarse (not very used in every day language).
In spanish the sustantive (Alegría/regocijo) or adjective (Alegre/regojijante?? lol) are better than verbs for domains. The best verb tense is "infinitivo"--> Alegrar or regocijar.
I talk in general, you can find verbs which works better.
hope it helps you.