travelling to Taiwan

You don't really need to book day tours. Taipei is very easy to get around as an English speaker. Buses and the MRT (metro) have signs in English.

Here are a few things that might be of interest to you.
Taipei 101 - At one time the tallest building in the world. You can go to the top if you want. There is a good food court and a shopping mall on the ground floors.
National Palace Museum - Filled with some of the best artifacts from the Forbidden City.
Chaing Kai Shek Memorial - It was always closed for "renovation" when I was in Taipei, but now that the KMT is back in charge of the government, it has re-opened. The grounds nearby also contain 2 large performance arts buildings that could provide some good photo opportunities.
Sun Yat Sen Memorial - While I wouldn't go out of my way to see it, if you're in the area it might be worth a stop. There is a small museum dedicated to him on the grounds.
Longshan Temple - Taipei's largest and most active temple.
There are some night markets that you might want to visit. Day trips to various hot water springs are also possible.

Sorry, I didn't spend enough time in Taipei to have specific eating recommendations. Starbucks is everywhere, but I don't know if you consider that "good coffee" or not (it's fine for me, but I'm not fussy).

Thanks for posting your info have been to the Summer Palace and so will be interested in seeing the artefacts from there and also intend checking out some of the temples.Am hoping the typhoons etc don't cause any more damage or loss of life.Have been to many Starbucks and don't mind their coffee even if it sometimes has to be a double shot! Did you visit any rural or coastal areas or go down by the river.

HERE are some ideas.

I used to post really long responses to this question, but lost the C&P that I'd use.

A little googling and reading through the blog index at Taiwanderful.com will produce a host of Taiwan-based bloggers who can give you an idea of where to go and what to eat.

My top Taipei favorites:

Dihua Street and Xiahai Temple
Longshan Temple and neighborhood (inc. the backstreets, lanes and markets as well as Guiyang Street)
Raohe Night Market and Wufenpu Fashion Street (for people watching, not buying)
Lin Family Garden and Mansion
Maokong
Tianmu Old Trail (go on a weekday early morning or late afternoon to see monkeys! - also good views of Taipei and a nice restaurant at the base near Zhongshan Rd.)
day trip to the mountains - tons to choose from. Pinglin, Daxi/Fuxing/XIao Wulai, Sanxia and Happy Valley, Yangmingshan, Bitou Cape...so many choices
Keelung Night Market
Bao'an and Confucius Temples
random backstreets and old temples of the Ximen area

My top Taipei eats:

that old mainlander Sichuanese restaurant in Dingxi

Raohe and Keelung Night Markets

Maokong's Mountain Tea House (next to Redwood tea house near the old gondola station) lemon diced chicken, sweet potato leaves, red-sauce stewed tofu and mountain pig

Ali Baba Indian Kitchen (north Indian)

Taipei Snow King (funky ice cream)

Hui Guan (Chinese Muslim food from Ningxia's Hui minority - Xinyi Road)

Yuanlin Rouyuan (Heping Fuxing Rd. intersection - Zhanghua-style rice gluten meatballs

Tainan Danzai Mian (southern Taiwanese food - Songde Road north of Xinyi)

Alley Cat Pizza (amazing pizza and tiramisu)

Yangon (Thai and Burmese in Gongguan)

Borneo - a little Indonesian restaurant in Shida night market

the famous shrimp roll guy and almond jelly shaved ice guy on Dihua Street - hard to find if you don't know what to look for

the beef noodle joint near Jingmei MRT exit 2 (handmade chunky noodles and very dark broth)

Cha for Tea - very light food, it's a chain - the food is made with tea

pretty much any beer and seafood joint on Chang'an E. Road, or one that I am fond of in Danshui

Sugar House shaved ice in Nanshijiao night market

Shanghai Dumpling on Minsheng E. Road east of Dunhua

the Korean restaurant in Shida near Salt & Pepper bar, one or two lanes west of Grandma Nitti's

most of the small snack vendors on Yongkang St. or in the Longshan Temple area

Food areas to avoid:

Xinyi
most of Dunhua Rd.
Ximending (really - you'd think the food would be good but it's not)
most of Shilin and Tianmu barring a few decent foreign eateries

Good coffee:

Naruwan Aboriginal Market (Guangzhou and Huanhe Rds. near Longshan Temple) - Taiwanese coffee grown by aborigines made in some fancy glass ball thingo...expensive but very rich and good

Almost any Vietnamese restaurant serving coffee

Black Bean Coffee (Zhongshan N. Road just south of Zhongzheng, MRT Shilin) - Monsoon Malabar Indian coffee

Wein Mountain Roast - Jingmei MRT Exit 2 - very overpriced but most blends are fairly good

Monument (Zhongshan and Chang'an Rds near Taipei Main) - it's in an old shophouse. Get their house blend. VERY STRONG but not bitter or charred. It'll leave you buzzed as heck.

I've been told Alley Cat pizza has good Italian coffee but I always drink alcohol there so I can't say

oh, and Red House Theater's Cho West Cafe - it's pretentious and expensive but they do good coffee.

Thanks heaps for taking the time to provide that info really appreciate it