Next, Tom remembers that he heard a doctor talking about something that takes two weeks to heal.
I have to identify what type of pronoun that (second one in sentence) is. Would it be a relative pronoun?
In this particular sentence, the first "that" is a subordinating conjunction introducing the dependent clause that follows it.
http://grammar.ccc.commnet.edu/grammar/conjunctions.htm#subordinating_conjunctions
The second "that" is a pronoun, and yes, it's a relative pronoun.
http://grammar.ccc.commnet.edu/grammar/pronouns1.htm#relative