Getting permits is simple. I just hiked Snow Mountain again two weeks ago with one other foreign friend. I applied about 2 weeks in advance online. It was very straightforward though you will need some help if you don’t read Chinese
Also, check park announcements for trail conditions and trail closings which can happen often in typhoon season. It is not a good idea to sneak into the National Parks on less-used trails. These are less used for good reason and you can encounter very difficult conditions. There is a long history of seasoned hikers from other countries getting themselves in serious trouble in Taiwan’s high country. Don’t join them.
As Mucha Man said, go during the week. It is really tough to get permits on the weekend because there is such demand. And it can get unpleasantly crowded on the main routes on the weekends. Lots of Taiwanese go to have fun with lots of their friends, so you can expect pretty boisterous company that bothers some people looking for solitude.
Yushan is easy for any one in moderately good shape. Snow Mountain is definitely harder (but still doable) but I think more beautiful in terms of the panoramic views from the top and the micro climates. Flowers everywhere when I was up at the end of June.
Might I suggest that rather than climbing one of the big peaks, you try the National Trail System? Far fewer hassles with permits and way less people. Just in Taipei’s backyard, for example, you can do a wonderful hike from Fushan in Wulai over the mountains to Lalashan on the North Cross highway. Cypress forests and hot springs in Lalashan await! Or slightly further afield, there are many more great trails in Hsinchu.
In Nantou try the Nenggao trail and hike along the central mountain range over the Hehuanshan.
Let us know if you need more information.