Hsinchu can be a harsh city to make friends in, especially if you don't look in the right places. So far, I've seen a ton of great suggestions. I'll add a few more.
For social drinking and dancing:
Red Bar located on Minzu Rd. 33-55, is a great place to meet foreigners, particularly English teachers.
Barfly located on Ren'ai St. and Zhongyang is also a popular destination for foreigner English teachers.
PLUR night club: On the corner of Beida Rd. and Xida Rd. is a good place to meet foreigners of both English teaching and otherwise, especially on the weekends.
If you are close to Science Park, check out N2 near the Royal Hotel on Guangfu and Jieshou, a ton of foreigner engineers go there.
Night in Tokyo Bar: Zhongzhang Rd, near the East Gate, across from the Mos Burger.
I'd recommend these places if you just want to go out and meet people (foreigners) randomly and have a drink.
Other than that, FB pages are your friend, such as Hsinchu Foodies and Hsinchu Sports Page. The largest Hsinchu FB page is Hsinchu Teacher English Forum.