You can check out my opening post to this thread for a link to the bike rental thread [url]http://tw.forumosa.com/t/sticky-unofficial-forumosa-cycling-club-2016-taipei-new-taipei/88731/1
For flat bar bikes that are geared for long distances, I would suggest the Nanjing Giant Shop.
If you want a road bike with drop bars, I would suggest either the Specialized store in Neihu, Buffalo shop and Giant shop by Da Dao Cheng Wharf. Do be warned that the Buffalo Shop and Giant Shop tend to have pretty old bikes that they do not really do too much maintenance on.
I can't really help you in trying to find a local cycling club that does Wuling trips. Many clubs do, but it will be tough to find them, you're best bet is to FB search or maybe ask the rental shops.
As for your route, the west side of Taiwan is nothing exciting, but to avoid lots of traffic lights and air pollution, I would highly suggest you stay on the coastal highway and to get it out of the way. Two days from Taipei to Kaohsiung is do-able, but might be rough on the legs. Spend more time in the southern peninsula/Kenting area and take your time up the east coast.
As for ending with Wuling, I highly suggest you take anything from full day to two days off of the bike in Hualien or Xin Cheng to rest. You will need all the strength you have to get up to Wuling. Of course it depends what kind of pace and distances you're riding at, but that's just my suggestion. FYI At a higher pace, most mortals make it to the top at around 7 hours from the Xin Chen area.
Make sure you pack a bike bag of some sort. If you plan on ending your trip in Puli (western side of Wuling), there are only buses out of that town mostly to Taichung or to Taipei. Buses will only allow your bike bagged in the lower compartment of the bus.
Feel free to PM me if you have any other questions. Have fun!