Having been to Taichung twice (Dec 2013 and Dec 2014), I would say that I am relatively familiar with getting around Taichung using public transport. There is, unfortunately no MRT/MTR/Metro/Subway system in Taichung City, so the only public transport services available are the buses and taxis. If you are really on a tight budget and do not mind sitting or standing on buses for a long time, take the bus. It is fairly cheap to take buses in Taichung since the first 8km of most rides are free. It might be all rides, but I am not too sure.
If you are taking the buses, do take note of the intervals and waiting time for each bus. Some intervals are really long, so it might be good to do some research to see what other buses in that calls at the bus stop where you are at or other bus stops near can also take you to where you want to go. The bus stops are not always sheltered, unlike what we commonly see in Singapore, so it might be hard to find one. Look out for poles and signages with lots of numbers, they might easily be the bus stop you are looking for. Check that it goes in the direction you want it to; some includes a route map and indicate the direction in which the bus is heading but if you are unsure, to check with the driver.
One useful resource is a website managed by, I believe, the city officials. It offers real-time bus updates (when a bus would call on a certain stop) as well as the route map. It is in Chinese, so it is best if you can read it. The bus are numbered, so you should still be able to find the bus route you want to check out by selecting it from the dropdown menu. Once the route map is loaded (with the Google Maps backdrop), you can click on the little bus stop location markers where a little dialogue/message box would pop up. Click on the button, it will give you a list of other buses you can take from the bus stop. (http://citybus.taichung.gov.tw/ibus/RealRoute.aspx)