Make friends in Taipei. Hopefully a girlfriend. So you can stay the night.
Do not make your six months here an ordeal. Hsinchu will keep you busy if you give it a chance. Explore the rest of the island, meet people from all over, and live confortably -and a much cheaper price than in Taipei.
Trains are efefctively connected here. Buses take you anywhere all over Taiwan, yes, from Hsinchu. Airport? That si what highways are for. No problem traveling around or abroad from Hsinchu. You do not have to be in Taipei to get anywhere but Taipei.
Taipei has an airport that only goes to japan and Mainland China. Same time from Hsinchu or Taipei to Taoyuan airport to get out of here, say, to Thailand.
Hsinchu has the science park, the college areas for partying. If not enough, then yes, head to Taipei, but you won’t be going out every day anyways. Or better yet, try Taichung, or head a weekend down South to Kenting or Kaohsiung. Hsinchu is a major train hub, you can choose between bullet train or normal trains. It is the same line as in Taipei, you will not miss a thing.