Carlton Hotel on Beida Road is probably one of the most convenient and comfortable hotel in Hsinchu. The hotel itself is a bit old style, but it's well maintained and quiet and comfortable. It's situated within walking distance from an old temple with night market, there are plenty of places to eat and hang out. You should try the buns filled with sesame and red beans. Try the local noodles.
You can try TRA (local train) to get to different places from Hsinchu station. You can even go to High Speed Rail station using TRA, though a little slow and they only have a couple of train rides available every hour.