I did the research awhile ago and using public transportation, it'll take you just about 1-1.5 hours to get from HsinChu bus/train station to the harbor. Probably over 2 hours if your starting point is the HSR station since that's in Zhubei. Life without your own vehicle outside of Taipei is very inconvenient.
If budget isn't an issue, taxi is the fastest option. If you get enough of your friends, the taxi fare shouldn't be too bad split among a few people. It's about a 15-20 minute drive from HsinChu station to the harbor. Main issue is you might have trouble finding a taxi when you want to leave the harbor. You might have to call 55688 or some other taxi company to come pick you up.
All the above is on top of the trip from Taipei to Hsinchu. An HSR ticket (I think) is around 300NT, but like I mentioned above the HSR station is located farthest away from the harbor. TRA (150NT ticket) and bus (125NT ticket) from Taipei Main you can take to Hsin Chu station and connect to local bus or taxi.