getting to Tainan is pretty simple with HSR. once you exit out of immigration, you should find a sign that says metro. Its basically going to the basement of the airport. You will find the direction to go towards the High Speed Rail Station.
Once you get to the HSR station, u can buy the ticket there, the agent usually will know the next train down and factor in your walk time to the station so no worries on that. I wouldn't recommend buying a ticket beforehand unless your really good at timing. Too many factors to worry about (ie delayed flight + timing from Airport --> HSR Station and etc). Although if you are travelling on a very busy season, then that may be a downside to it as u may need to stand if no reserve seating is available. When you say first class, do you mean business class seats? If that's the case, there should always be seats available (normally). also there is a dedicated line if you don't plan to use the machine to purchase your ticket.

As for luggage, the only issue u may have is just pushing it around and going up and down the escalator/elevator. you may have to hold your baggage on the metro if the luggage rack is full.