As many have said the "proper" way to refer to the currency of Taiwan is 新台币 (Xīn tái bì) which is literally broken down to 新 (Xīn) = New and 台币 (tái bì) = Taiwan Dollars
Old Taiwan dollars are referred to as 舊臺幣* (旧台币) (jiù tái bì)
However you would only refer to them by these proper names when dealing with multiple currencies. When referring to local currency you would use 块, 毛, 半 (Kuài, máo, bàn).
Refers to 1 unit of currency relative to where you are. So 一块 (Yī kuài) in Australia refers to $1 AUD, in Singapore refers to $1 SGD, in the UK refers to £1 GBP, and in Taiwan refers to $1 TWD.
Now because you're in Taiwan you won't be using much more then 块 Kuài because the half dollar is rarely used. But please have a read since you seem to travel a lot and it will be helpful to know:
Refers to 10's of cents so 一块五毛 (Yī kuài wǔ máo) would be $1.50 in AUD, $1.50 SGD, £1.50 in GBP, etc ...
Now 半 (bàn) isn't a unit of currency but a modifier whose value is equal to half of a single unit of its parent value. Now that seems a little confusing at first but it's the same as saying "and a half" in English. So let's move on to some full prices of some items:
六块九毛半 (Liù kuài jiǔ máo bàn) = $6.95
一百八十九块半 (Yī bǎi bā shí jiǔ kuài bàn) = $189.50
Note: 块毛半 (Kuài máo bàn) only works for currencies that use whole values. Under the older systems of currency like the old British system of 12 pence to a shilling, 20 shillings to a pound this would not work and you would have to use the formal names of each unit.
20块新台币today. The local inhabitants of Chinese origin in Malaysia and Singapore also say
块. Taiwanese reformed their currency in 1970s or so, and thus the current currency is also called 新台币.