If you find yourself in Hsingchu and are on a budget you can find some busses that will take you around for free. They are a little limited in where they go, and sometimes, how often. But if you are a student you know that the free option is always a good option! And if you are in Taiwan you know that the option to enjoy an air condition might be conviction enough to get onboard.
There are some free shuttle busses who are driving from and to the Universities in town, if you have a patience to wait some time for the bus they are totally OK.
If you want to visit the shopping malls in town they pick you up on the train station and departures really often. But as soon as you get off the bus at one of the malls you will realize that everything around is branded and quite pricy. After spending some time walking around and mostly finding stuff we could have bought in Sweden (and sold for the same price) we went outside and got a cheap but filling dinner from a vendor. As least we got to spend some hours inside without suffering in the heat!