You would say, "there are a lot of people". In speech, "there is" can become "there's", so it's possible to hear speakers use "there's a lot of..." when they should be using "are".

Hi Rose, I hope I don't give you too much grammar here, but I feel that you might be having a problem with the (existential) ‘there’ at the beginning of your sentence. 'there' is in the usual subject position in the sentence and for that reason it is often called a dummy (pretend) subject. How do you make subject-verb agreement with ‘there’ in subject position? The answer is you need to look after the verb for the subject – in this case ‘a lot of people’ and decide whether this is plural or singular. Since ‘a lot of’ can be used with both non-count nouns (food, money, water), or plural count nouns (cars, people, children), you need to ignore ‘a lot of’ and focus on the head noun, in your case, ‘people’. ‘People’ is a plural form, and you cannot say ‘people is’ so your sentence should read ‘There (dummy subject) are (verb) a lot of people (real subject)’. Hope this helps :)

No, you have to say "there are..." if what follows is plural. In casual conversation, native speakers often contract "there are" to "there's". This is technically wrong, but still very common.

No. There ARE a lot of people.

Only THERE ARE a lot of people