Are there buses / trains from Taoyuan Airport to Hualien and from Hualien to Taichung?

Yes.
From the airport to Hualien, you can take shuttle bus to Taipei MRT Station. Then many trains can take you to Hualien. The travel time is 2 to 3 hours.
From Hualien to Taichung, you can also take the train at this station. It takes around 4 hours.

If you want to go to Taichung from the airport directly, you can take shuttle bus and arrive at Taoyuan MRT Station. There are several trains that can take you to the destination.