Wat you need to do is use the SBB planner. Just use www.sbb.ch and enter “Zurich Airport” in the from box, and “Villingen” in the to box. You will get different routes proposed, and you will see that you do not always have to go to Zurich HB first, as you can also go via Konstanz.
HB stands for “Hautpbahnhof” or, in other words “Main Station”.
The different train indicators and numbers are just there to help you find our train.
The railway station is in the airport, just follow the signs, and of course you can take luggage. Trains wouldn’t be very useful otherwise, wouldn’t they?
Since you are arriving by plane best thing to do is just buy your ticket at the station, and then the ticket clerk can explain everything to you.