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Spanish is a bit like Latin in that verbs have fairly complex conjugations and there are several basic conjugations that follow standard rules which, as you fear, have to be memorized. There are three basic conjugations for regular verbs. And then there are oodles of irregular verbs.
Both "comimos" and "fuimos" are past tense. There is a pattern you can follow to form the conjugations, but keep in mind that some verbs (many common ones) are irregular and will vary. This is the model for regular -ar, -er, -ir verbs in the pretérito perfecto simple:
yo -é -í
tú -aste -iste
él/ella/usted -ó -ió
nosotros -amos -imos
vosotros -asteis -isteis
ellos/ellas/ustedes -aron -ieron