I didn't want to answer this question, but I feel that I have to because I think you've been given a couple of misleading and false responses.
Your sentence should be:
"The first thing I do in the morning after getting up is brush my teeth and wash my face, but the first thing he does is go back to sleep."
If you want to use a semicolon, go ahead:
"The first thing I do in the morning after getting up is brush my teeth and wash my face; the first thing he does is go back to sleep."
The second sentence is grammatical and perfectly acceptable. It's not what I'd call normal idiomatic English only because most native speakers don't use semicolons unless they absolutely have to. I don't blame them for that, but I use them all the time; however, I'm a technical editor, so my daily writing context is almost always different from the one that your sentence suggests yours probably is.
There are a host of websites with "rules" about using semicolons. The rules they give aren't all the same. These two, however, are relevant to your question, and they're consistent and believable:  Use a semicolon to [connect closely related ideas], and  "A semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought." This link gives an example sentence that parallels yours: "Some people write with a word processor; others write with a pen or pencil." If you wanted to replace the verb write with a comma, you could: "Some people write with a word processor; others, with a pen or pencil." They're both grammatical and structurally parallel.
Your final sentence, the one that includes "...the first thing he does, go to sleep.", is grammatically incorrect. It has to be "...the first thing he does is go back [NB: EDIT: Corrected tense mistake pointed out by Tim Lymington's comment. Sorry about that oversight.] to sleep." You can delete back if "he" works every night and sleeps during the day, however. But if you mean that he wakes up after a night's sleep and then returns to sleeping instead of getting out of bed, you should include back.
If you want to take Kurt Vonnegut's advice, Here is a lesson in creative writing. First rule: Do not use semicolons. They are transvestite hermaphrodites representing absolutely nothing. All they do is show you've been to college., no one will criticize your writing for it.