In Spanish, there are two words for "a lot". And no, they are not entirely synonymous. Muy is an adverb:
  • Adverbs qualify verbs (to run, to talk, to play)
  • Adverbs qualify adjectives (nice, big, tall)
  • Adverbs never change
Juan es muy alto.
Juan is very tall.
Mucho is an adjective:
  • Adjectives qualify nouns (the table, the ball, the singer)
  • Adjectives change to match the nouns they precede:
One last little thing: You can use mucho after verbs!
Julia trabaja mucho.
Julia works a lot.
Mis padres viajan mucho.
My parents travel a lot.
Ella habla mucho.
She talks a lot.