You should look up that word in the dictionary.
What is the common way to say this in German?
Du sollst das Wort im Wörterbuch nachschlagen.
Which verbs are acceptable? I'm thinking of nachschlagen, nachsehen, nachgucken. Would "anschauen" work?
You first set of suggestions is correct and only missing karoshi's addition of
Ordered by register from high to low:
nachschlagen, nachschauen, nachsehen, nachgucken
anschauen is not correct. It would mean, to look at the word as it is printed on that very sheet of paper or at the blackboard or whereever.
I would use "nachschlagen". "Nachsehen", I would more think of not counting an error against someone or in the sense of "extenuating circumstances", e.g. in "he is still a child". "Nachgucken" has for me the connotation of look after a beautiful young woman that just passed past me. "Anschauen" is way off and can not be used if you want to say that you look at something like a nice painting.
To summarize, you could probably use the first three words, but the nachschlagen would be the best to use and nachgucken the worst (I consider the use of "gucken" as unprofessional). Anschauen is not an option.
Let me summarize the meaning of each of the proposed words as I feel them as a native speaker: