English question, please help. Thanks.?

I think the second one is correct!

I think it should be, "There's a lot of stuff that I need to do". "Is" and "a lot" both are singular and describe "stuff" and "stuff" is referring to one group of things.
I'm pretty sure that's correct, but I live in America and I hear a lot of slang everyday and I sometimes use it without knowing.....

I think that grammatically the second one is correct. The first one would have to say "There's a lot of stuff that I need to do". Personally I think nobody would notice if you say them either way. As far as writing a paper or anything to be critiqued, use #2.

Oddly enough, I believe the first one is actually the correct form. I believe this is because stuff is always singular, so you always use a singular verb with it. Most likely, you're looking at "lots" and thinking, "oh, this is plural," but lots isn't actually the subject. There ARE lots of things and there ARE lots of monkeys, but there IS a lot of stuff, because things and monkeys are plural, but stuff is singular. I've never heard anyone say "there are lots of stuff."

There is a lot of stuff that I need to do.

Lots is slang....I'm guessing you want grammatically correct