"They are trying the wine."
Translation:Δοκιμάζουν το κρασί.
I think that would be like "they attempt the wine" :)
I would use προσπαθώ for trying to do something (making an attempt) but δοκιμάζω for something closer to "judge", i.e. take a small piece to see whether you like the taste, or even trying an activity to see whether you like it.