I'm unable to figure out if the following sentence is correct:
"After I watched the movie, the first thing I did was shutdown the laptop and go outside"
Is this grammatically correct? Or should it be "..went outside". Please help.
The only thing wrong with the original sentence is that shutdown is a noun. You are looking for shut down, the verb.
The rest of the sentence is fine. Your confusion stems from the fact that shut is both the bare infinitive and the simple-past form of to shut. With a different verb in its place for which that is not the case, it likely wouldn't have arisen.
Constructions of the form "[what X] [did was|do is|does is]" take the infinitive. Here are the actual usage stats from the Corpus of Contemporary American English (COCA) and the British National Corpus (BNC):
COCA BNC did was [infinitive] 726 138 did was [simple-past form] <10 0
As a side note, don't listen to people who insist (even on this page) that the infinitive must not be a bare one. Quite the opposite is in fact true:
COCA BNC did was [bare infinitive] 476 47 did was [to-infinitive] 250 91
That is, both forms are fine, and the to-infinitive is preferred by a factor of less than 2 in British English, while for American English, the preference is actually flipped. So, as is often the case with language, it doesn't hurt to know your audience.
For further information and even more stats, see e.g. “All you have to do is read” vs. “All you have to do is to read” and the many related questions linked from there.
"The first thing I did was to shut down the laptop and go outside" is right. You are not supposed to use went if the sentence is already in the past (I did).
You can take that like a bit of brain logic :)
The first thing I DID
gets you to a PAST reference
was TO shut the computer down ...
the Infinitive keeps you the same level or logical plane (you name it) as the main PAST you have in DID.
You can 'stay there'
and GO out ...
I hope this can help. :)