Which is correct?
I would like to go to a youth center to help and work with teenagers.
I would like to go to a youth center to help and to work with teenagers.
The question is whether to needs to be repeated in the second infinitive.
I would use the second infinitive, to clearly convey what I think you want to convey.
The issue I have with the first example,
is that, because of the undivided phrase "to help and work", structurally speaking, you're saying two things:
And I don't think you want to say that you are "helping with" teenagers. That makes it sound like you're helping a parent handle their unruly teenage children, as opposed to actually helping teenagers themselves.
If you use the latter construction, dividing the phrase into separate infinitives, it makes it clear that you want to help teenagers, and that you want to work with teenagers.
A simple test to see whether you should use a second "to" in a sentence is to switch the order of the words connected by "and":
... that sounds horrible, so keep the second infinitive.
An alternative solution would be to say
though personally I think that sounds awkward.
Since "to work with teenagers" is a form of "helping", I would say you are safe to omit the to.