Translation:Please wear a shirt, and then put on a tie.
I feel like 'Please wear a shirt and put on a tie' is more natural English, the 'then' to me kinda implies the finishing of an action, like 'please wear a shirt and then [once you've worn it] put on a tie'. Well, the most natural would be 'Please wear a shirt and tie' but that's not the literal translation so would be harder on non-native English speakers...
"Please wear a shirt and then..." is very strange English no matter what follows. The "and then" with a continuous action like that implies that you're supposed to finish wearing the shirt before putting on the tie (that is, you're done wearing it and take it off before putting on the tie). That's a strange thing to ask someone. :) Of course, people will understand what you mean, but it will sound strange.
We could use a similar sentence structure with more appropriate actions: "Please go swim in the lake and then dry off with this towel" is a more reasonable request (if a little presumptuous). You're supposed to be done swimming before you dry off.
Without "then" it would be okay. "Please wear a shirt and put on a tie" is reasonably natural.
"Please put on a shirt and tie" or "Please wear a shirt and tie" are also natural. There is a bit of a difference in meaning between the two. With "put on", it's like you're asking the person to change their clothes (either right away, or at some point). With "wear", it sounds like you're telling them what would be appropriate for some sort of occasion, a party, or work, or something like that.
"Please put on a shirt and then put on a tie" is mechanical and a little strange, but it means the same thing as "Please put on a shirt and tie" -- it just sounds like instructions given by a robot to someone in a psychological experiment or something.
それから ("and then") is used in other sentences to describe two actions in sequence. I've gotten this wrong in both Japanese and English because it appears in the English translation but not in the Japanese. It should either be "Please wear a shirt and a tie" or 「シャツをきて、それからネクタイをしめてください。」