There are really three kinds of of drinking/going out places. Taiwanese drinking-restaurants, high end cocktail bars and a handful of disco-bar/nightclub places.
Taiwanese people, if and when they drink, generally do this with food. A place like this isn’t hard to find. Normally there’s a neon beer logo outside and you will see big round tables with decently-sized beer bottles and large dishes surrounded by laughing and chatting families and workmates.
These places are rowdy and convivial, so going there alone is not recommended. Since you’re eating and drinking you might spend more than planned, on average a dish is 70NT+ ($2.20) and a bottle of beer is 100NT+ ($3.20).
For a more westernized experience, around the 101 and other affluent districts there are plenty of very luxurious cocktail bars and coffee shops which have all the drinks and atmosphere of an equivalent bar in New York or London. At a place like this, drinks start from about 300NT+ (10$) but they can easily cost more than 600NT ($20).
A good place to have a more sophisticated evening is Huashan Culture Park, between Shandao Temple station and Zhongxiao xinsheng station. It is a renovated Japanese factory complex that not only has a couple of bars, it also has an excellent pizza restaurant (alley cats) and numerous exhibition spaces and venues. A great way to spend a Saturday evening with a special someone.