Muy vs. mucho
While both muy and mucho can relate to the idea of a lot in English, they are not interchangeable in Spanish. The key is understanding what kind of word they are modifying.
Muy
A1Muy is an adverb that means very or really. It is used to intensify the meaning of an adjective or another adverb. Muy is invariable, meaning it never changes its form.
El café está muy caliente.
(The coffee is very hot.)
Mi hermana es muy inteligente.
(My sister is very smart.)
Tus amigos son muy divertidos.
(Your friends are very fun.)
El tren va muy rápido.
(The train goes very fast.)
Vivimos muy lejos de la ciudad.
(We live very far from the city.)
Ella canta muy bien.
(She sings very well.)
Mucho
A1Mucho can be an adverb meaning a lot or an adjective meaning much, many, or a lot of. As an adverb, it modifies a verb and is placed after it; in this case, it is invariable. As an adjective, it comes before a noun and must agree in gender and number with that noun, changing to mucha, muchos, or muchas.
Hoy he trabajado mucho.
(I have worked a lot today.)
Mi abuela me quiere mucho.
(My grandmother loves me a lot.)
No tengo mucho tiempo.
(I don't have much time.)
En la fiesta había mucha gente.
(There were a lot of people at the party.)
Tengo muchos libros en mi habitación.
(I have many books in my room.)
Ella tiene muchas ideas buenas.
(She has many good ideas.)
Summary
In short, use muy (very) before adjectives and adverbs. Use mucho (a lot/much/many) after verbs or before nouns. Remember that when mucho is used before a noun, it must change to mucha, muchos, or muchas to match the noun's gender and number.