I'd go for Pronoun because it is an unspecified place. You could be going anywhere ... as long as it is warm there. Compare this to "I met him somewhere." Somewhere here is an adverb because you met him at a specified place (you just can't recall where, but it was at a specified or certain place.) Saying all the above, there is some variation on how grammar books describe "somewhere" (pronoun/adverb/something different). So it'd be an unfair question if it were in an exam. A lot of grammar books aren't in agreement.

I'm not a grammarian, but I think it is a noun. You can reduce the sentence down to just: "I would like to go somewhere". 'Somewhere' is the (indeterminate) place that you want to go. Although dictionaries abbreviate parts of speech, we don't usually do that in writing, so I suggest you avoid that in your questions as well.

The word 'somewhere' could be an adverb, but here it is a noun.