Welcome to Taiwan. It great that you’ve come to see family and also experience Taiwan firsthand. Here’s to you finding some answers.
You can also buy train tickets ahead of time conveniently in 7-11 at a machine and the store clerks are usually really friendly and helpful if they have a minute. Many people in Taipei speak English.
You need to buy the ticket for the date and time that you want to use it to get a seat. I’m going to be leaving for vacation on Wednesday otherwise I’d help you guys get around the train station on Thursday. I’m sure there are other Forumosans that can help.
If you and your friend would like to meet for a coffee or beer Monday or Tuesday, send me a message. Best wishes!