I was there in April/May for the first time in a few years. There's a lot to do, especially around the Taipei area if you don't have a car, reachable by MRT and a bit of walking. You can also hire a taxi for the day for day trips, etc. (I think it might cost around $100 US).
I think we had a thread about something like this a few months ago?
There are quite a few ex-pats-in-Taiwan type blogs, etc. Here are a few I've bookmarked:
http://www.forumosa.com/taiwan/viewtopic.php?f=55&t=16531&sid=bb96505219dfe8cde44e0678c8fc98cf
http://forumosa.com/taiwan/viewforum.php?f=55&sid=89886d1a1880cee23e969bdb66a22164
https://hikingtaiwan.wordpress.com and the old site:
http://hikingintaiwan.blogspot.ca
http://www.darrenmelrose.com/blog/index.php
http://www.craigfergusonimages.com/#!/blog
http://www.photojazz.ws
http://formosaguy.blogspot.ca
http://michaelturton.blogspot.com
http://www.thenhbushman.com
http://blog.taiwan-guide.org (check out the Taiwan blog-roll links)
https://www.flickr.com/groups/taiwanphotoclub/
http://taiwan-photography-blog.com
I did day trips from Taipei mostly, but if you can, a trip down the east coast is definitely worth it, as well as venturing in to Taroko Gorge. I did a cross-island drive through the mountains that was pretty much the highlight of the trip. It was also a great reprieve from the heat of the cities. But you'll have to check on current mountain road conditions considering some roads were damaged from the last huge typhoon back a month or two ago. The main cross-mountain roads should be OK, but some of the smaller ones might not be. Kenting at the very south is nice too, if you want a bit of a resort vibe. It reminded me a bit of Hawaii. Taking the high speed train from Taipei to Kaohsiung could also be an option for getting a quick west coast snapshot of the country. Pricing is fairly reasonable and you could even do it as a day trip. If you're into it, be sure to check out a hot spring. There are lots of them!
I still haven't processed my images from the trip this year! Just backlogged with jobs, etc., but here are some of my photos from Taiwan back from the mid 2000s...
http://www.ronscheffler.com/blog/?cat=10
I like hanging out in the cities, particularly Taipei because it's easy to get around with the MRT, and people watching, wandering down side streets, visiting temples, parks, etc...
A couple popular spots near Taipei you can reach easily by local train are Pingxi and Jiufen:
https://en.wikipedia.org/wiki/Pingxi_District
https://en.wikipedia.org/wiki/Jiufen
Pingxi is known for its lantern festival, but you can buy lanterns any day, write a message on it and send it aloft. I like it for the train ride, which I take to the end station (Jingtong) and then hop back through the small towns (buy a day pass at Ruifang station) and/or check out Shifen waterfall.
Jiufen is a neat hillside town. A bit touristy, but still nice. You can get there easily by bus or taxi from Ruifang.
https://goo.gl/maps/rjW4TLGG8Mz
Hope this helps...