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
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!