Din Tai Fung is traditional - shao long bao (aka soup dumplings). There are four branches in Taipe, and they are always packed. They are well used to handling tourists.
Shao Shao Ke is my favourite restaurant - delicious northern Chinese cuisine in a very funky restaurant. Reservations needed on weekends but not usually on weekdays. I strongly recommend the shredded potatoes with chili and vinegar, and the steamed/baked/deep-fried pork rib. The bread soup is good too. English isn't necessarily spoken, but the menu is bilingual. I would advise making reservations, and ordering the fried cheese dumplings for dessert - they have to be ordered in advance, but are worth it.
Definitely plan on a night market dinner. Shilin is the biggest, but the Raohe market is good if you want something more local, less touristy. Basically, stick with cooked and popular and you should be fine. My favourites include the spring onion pancakes, grilled sausages, deep fried chicken (aka ji pai), barbequed squid on a stick, oyster omelets and of course stinky tofu, which tastes much better than it smells. Unfortunately, it smells like a sewer problem from a block away.
For other restaurants, for a light meal there's a neat place on Yongkang St called Tu Hsiao Yu which sells handmade Tainan style pork noodles which are quite tasty. You can finish off with dessert at Ice Monster down the street - shaved ice with fruit and condensed milk. The mixed fruit plate is good, even though the mangos aren't in season at the moment.
For an interesting hot pot meal, there's the Mushroom Park restaurant which does mushroom themed hot pots - (with mushroom appetizers). Very tasty, but little English.
Had an early afternoon snack at Tu Hsiao Yeh on Yongkang St (one of 3 locations in Taipei), and noticed they changed the menu. The chawanmushi is no longer being offered (sob sob) but the menu is now in a very readable format complete with photos and text in both English and Japanese and I believe I saw some Japanese characters. There was a short wait even at 2 to 3 pm! The noodle soup was as good as ever.
And just around the corner at the HQ of Din Tai Fung, the line (or lack thereof) was crazy, people literally piling up into the street, must have been 40 to 50 people waiting. I remember going for the first time in 2002 and there was no wait. And today is a Monday...
There is a beef noodle festival in Taipei every year, and in fact, the voting will start soon for the 2009 festival soon: http://www.2009tbnf.com.tw/
Unfortunately there is not much information on the 2009 website yet in English (if you can read Chinese then lots of info there), but I found this English list of "60 beef noodle restaurants" from the 2008 website: http://www.taipei.gov.tw/cgi-bin/SM_t...
There are so many other kinds of great food in Taipei, and not just beef noodle soup though.
I missed the beef noodle festival which took place in mid to late November, but if you visit some of the tourist offices, like the ones in the metro mall (underground shopping area) by Zhongxiao Fuxing station, they still have the 2009 beef noodle soup festival brochures. They do not mention the winners, but they list many recommendations as reviewed by the "Taiwan Michelin" guide, comprised of a panel of 12 judges who are self proclaimed beef noodle experts.
Now I withdraw my previous recommendation of Lin Tung Fong. While they are very good for a first time visit, it turns out there are much better versions.
The festival website mentions the winners of the beef noodle festival competition, but they rate and rank them by category. First is best clear broth stewed version, 2nd is hung shao or soy sauce and perhaps spices/chili oil stewed, and third is called creative category. What makes it more difficult to find out which restaurant won is the fact that the winners are listed by name of the chef.
However after a lot of searching around, the winner of the hung shao niu rou mien category is a chef whose last name is Wang, who owns a place called Liu's that specialies in sour cabbage and pork belly hot pots and of all places in the south in Kaohsiung (not Taipei). A brief bio mentions chef Wang is retired military and has a very straightforward but committed receipe. Can't recall who was in #2, but it was a Northern Chinese noodle and starch specialist place (Bei Ping Guan or something like that). Now #3 runner up is interesting...a Persian guy named Davod who has been living in Taipei over 15 years and with his Taiwanese wife, run a place called Lao Wai Yi Ping Niu Rou Mian somewhere near Yuanshan station (although the staion before that might be closer). He fuses some secret herbs he imports from Iran and makes a more robust stewed fusion version of the Taiwanese hung shao beef noodle soup.
Of the clear broth category I only researched the first one. The winner is a female chef and they say she is the owner's wife. The restaurant they own is a place called Mala Tien Tang (numbing spicy heaven). And on top of that, their specialty beef noodle soup was only offered in March this year. The wife is from Yunnan China and they use a special clay pot stew for the beef noodles, normally a prep they use for clear herbal broth chicken soup (Chi Guor). This is a very interesting and unusual prep.
While I did not try any of these thus far, I did hit a place called 72 beef noodles has a stellar version of a Shandong style baitang (white soup) where 1.5 kilos of ox bones are cooked for 3 days to make one serving of soup. Hardcore delicious. No MSG. They also use Himalayan rose salt for enhancing the taste. Their other stir fry plates are great. This place won 4 stars (highest award) by the "Taiwan Michelin" guide.