At I was visiting Cingjing and Jiufen, I rented a SUV (first time driving SUV) for 3 days just for the experience of driving LHD and covered the following routes:
taipei(ximending) > jiufen > taipei(maokong) > taipei (shihlin market) > taipei (ximending)
taipei > cingjing
cingjing > taoyuan airport
Maybe it differs from person to person, but for me, a GPS is a must to navigate the complicated roads in taipei city and the various highways, upramps, downramps, flyovers in and around taipei city. I would be hopelessly lost with it.
Taipei traffic is generally heavy and carpark is not easy to find, I had to park at the public multistorey carpark about 10 min walk from my minsu.
For the places that I went, the road are properly paved and are in very good condition, even up the mountains. Of course, you have to pay toll to enjoy it.
Somehow I managed to take it easy and enjoyed it and feel that it is not too stressful.
It would be good to drive in Cingjing as some of the tourist spots are just that much easier to get to by car. It is a mountain so to avoid walking up and down it, a car is definitely a good choice.
One thing to note though, the GPS that I got was taiwan chinese input by default and it is not hanyu pinyin. Fortunately, it does support handwriting recognition.
Also, being kiasu and avoid time wasting while driving, I also spent some time to research the routes, parking spots and also read up on the traffic rules and driving habits and etiquette in Taiwan.
Of course in the city itself, driving is definitely not a must, e.g. I used the mrt and bus to get to the various attractions such as Taipei 101, ronghe night market, wufenpu, national museum, danshui etc
Ultimately, the purpose of driving is for convenience and freedom to come up with itinerary that suits you. Have fun!!!