I'm not a teacher, but I would say this: The correct phrase, in terms of verb-number agreement is "there are a lot of people". What you will often hear, and slightly less often read, is the contraction "there's a lot of people". Even here in Zild, not renowned for its scrupulous adherence to formal standard English norms, the phrase "there is a lot of people here" would be uncommon. It seems that when the grammatically incorrect contraction is expanded, the grammatically correct form normally takes its place.
I am not a teacher but it is my hobby to teach English to young boys who are interested to improve their knowledge and speaking ability. Here the question is whether a singular form of verb "be" will be used with the expression "a lot of". This expression is prepositional phrase which can be used with both countable and uncountable nouns. If the object of the expression is an uncountable noun then singular form of the verb is used. Example:There is a lot of tea left in the cup. (tea - uncountable)
On the other hand if the object is a countable noun, the plural form will be used. Example:There are a lot of pictures on the wall. (pictures - countable)