I will probably hitchhike but it is winter and rain is not unlikely.
There are lots of discussions about getting from Taoyuan airport to Taipei, but I've actually been staying in Taoyuan for a few days. I'm quite close to Taoyuan station.
I don't care about speed or efficiency. I'm in no hurry. I'm just looking for the cheapest way to get to Taipei. I'm guessing it would be the regular (non-HSR) train, or perhaps a private bus company.
If it makes a difference I'll be heading to the area around "Small South Gate".