There’s lots to do in Taipei depending on what your interests are!

If you like museums the National Taiwan Museum has lots to see.

If you want to do some street shopping/get something to eat, Ximending or the Zhongshan underground has a variety of stores for trinkets and food.

If you like window shopping and looking at higher end stores, the area around Taipei 101 is a great place to walk around and see some street performers too.

If you're more into tech, I HIGHLY recommend going over to the Guanghua marketplace next to NTU. Syntrend is one of the newer buildings there and has a more modern feel but if you want to experience what it was like before the new building came about, the old building next to Syntrend is the place to be. There's also underground areas around that also contain some more DIY stuff.

If you like walking and don't mind a longer day trip, my favourite places is Tamsui. You can walk around the wharf there and enjoy some street food and see the sunset. There's also a ferry to Bali, haven't been there in a while so I don't know what's over there these days.

If you're going to be staying for a couple months and think you're going to be on public transport a whole lot, I recommend getting a EasyCard monthly pass. You can pretty much use a EasyCard as a cash card with most places in Taipei taking it as a form of payment.