Oct is the time where you can go to most places and Taiwan is a place with so many things to do!
Your plane will drop you at Tao Yuan international airport, which I assume should be around evening time. From there, the nearest you can go to is Taipei or if you are still early, head straight to é«é and subsequently to å¢¾ä¸ - which has beautiful coastal line as someone mentioned above).
The other option is to stay in å°ä¸ (cheaper than Taipei). From å°ä¸, you can go to æ¸
å¢ (at åæ) and SunMoon Lake (via åéé® from æ¸
å¢). æ¸
å¢ is worth staying at least for a night.
You can also go to é¿éå±± (from å義). Stay there for a night to catch the sunrise!
You can also visit Taroko Gorge (太é¯é£).
If this is the first time you are visitng Taiwan, you may want to spend 1-2 days in Taipei towards the end of the trip. In Taipei itself, you can travel to ä¹ä»½ and/or å å, æ·¡ æ°´ - these are old towns and street, good for photography. If you like shopping, try äºåå (for ladies) or 西éçº. And of course, don't forget to visit the night market and try out the different food!
Try to google for images of these places and see if you like what you see, then plan accordingly