Taipei & Taiwan

Yes, we were on a cruise that spend two nights there. We had two full days to tour.

One thing that we VERY MUCH enjoyed was our visit to the museum in Taipei. One of the best museums ever. Apparently, when the Nationalists were losing the war to the Communists, they took most of the items (except the large items) from the Bejing Museum to Taipei. It is amazing. One item is special, it is a massive tapestry called the 1001 horses, made by an Italian what was in China 300 years ago.

We also visited the massive National Chiang Kai-shek Memorial Hall in Taipei. We could have spent less time there, the hall is huge. It dwarfs the Lincoln Memorial in DC. Also, north of Taipei there was a seaside city with nearby amazing rock formations. I can't remember the name, but it was interesting.

The TripAdvisor Things to Do menu has everything sorted for you regarding day trips and tours based in Taipei.
This one might be of particular interest since you have a week in the country: 5 Day Best of Taiwan
In general, moving around the city of Taipei to see the sights is pretty easy using the excellent subway system - your hotel can provide a map and directions on how to use it. English signage isn't the best but it'll be adequate for your purposes. Can also use taxis for door to door service.
It's a fascinating culture with friendly people, and English is commonly spoken by anyone working in the visitor industry.

thank you for the insights thus far! I will look forward to more bounty!

Another tip as I think of it: always carry a business card with the name and address of your hotel in Chinese - will guarantee that you can always find your way home if your taxi driver doesn't speak English. Your hotel will have a stack of these things either at the reception desk or the concierge.