Taipei is essentially a condensed version of Tokyo. It has awesome street food, cool electronics, beautiful clothes and it's easy to get around via the metro.
Things to see and do in the city:
Shilin night market for cool souvenirs (starts at shilling station on the red line)
Chiang Kai-Shek Memorial Hall
Dalongdong Baoan Temple
Longshan Temple (oldest temple in Taipei)
Ximending for an awesome vibe, cool eateries, street art and stores (it even looks like a mini Tokyo)
Xiangshan (Elephant mountain) for a sunset view of the city
Raohe night market (Songshan station on the green line) for excellent street food
Tamsui to see the water front and for some excellent sea food
National Palace museum
Things to see and do just outside of the city:
Yanminghsan National park for a hike (Get a metro to Jiantan station and then Red 5 bus to Yanmingshan)
Jiufen Old Street for a beautiful view and to see the hustle and bustle of old Taiwan (Get a metro to Zhongxiao Fuxing on the blue line and then get bus number 1062 to Jiufen- the bus will take 45 minutes)
Food and drink to try:
Siwu Tang soup, Sesame oil chicken soup, Pork rice, breaded cuttle fish, Oyster omelette, Noodles OF EVERY KIND!
Bubble tea, Stinky tofu, Pigs blood rice cake, Shaved ice mountain, Pork bun, Dumplings
How to get around:
Taipei's metro system is excellent and pretty much always on time. It’s the most reliable way to get around.
If you want to visit a city where you feel completely safe, food is excellent and people are friendly then this is the city for you. Not only is it a great city for backpackers but it is also suitable for couples and families who just want to get away for a fortnight.
Is it truly budget friendly?… well, after southeast asia, no. But it is still relatively cheap. Accommodation being around £6-10 per night, food being around £2-3 per dish and the metro being around 50p- £1 per trip. Accommodation will be your biggest expenditure here.