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.