I'm familiar with the concept of how たくさん、多く、and 多い are used in relation to nouns (多くの人、人が多い、etc.) but I'm still unfamiliar with how you can say the expression "a lot of [verb]". I know, for example, たくさん食べる means "a lot of eating," but I'm unsure if you need to distinguish 多く and 多い when it comes to verbs.
In addition, here's what I know; tell me if I got anything wrong.
多く and 多い can carry an additional meaning of "too much." たくさん doesn't.
While there's a grammatical distinction between 多く and 多い, たくさん doesn't need this.