Yes, taxis are cheap in Taiwan, very convenient, and fleets of them abound around metro stations, hotels, and designated spots, not to mention doing the rounds. From the airport, you have many options in terms of limo services and taxi services. There is a designated waiting area for taxis, they are all metered, and the ID of the driver is clearly stated everywhere on the cab in case you have any complaints.
In any case, for extra safety, make sure you use a taxi from one of the many reputable companies around, meaning, better than an independent -more resources, more coverage, more service options. For example:
Taiwan Taxi - usually covered up in ads, and their phone number is 55688
Airport service -as stated, that's its name and duty
Tafong - blue ad on top
Women and children service -but not exclusively, recognize by the flowers they carry in the front
Moreover, I am constantly fighting with fellow foreigners who tell me they get dirty or weird taxi drivers. Finding a taxi in Taipei is easier than scratching your head, no need to get in the first one that follows you around. There are fleets with Mercedes, new Toyotas mini vans -the most popular, even fancy BMWs or ex race cars converted to taxis. So if you do not like it, just say no, and pick the one you want. Like MRT trains, that come every 3 minutes, taxis are one after another. No need to hurry. If you open a squeaky door and a blast of liquor and pinglang greets you, do not get in.