You will not find a lot of information about learning Dzongkha online….however, you can find a couple of sites that offer some information. Dzongkha is related to Tibetan and the script is very similar. Take a look at these sites, if you are interested, and try out some Dzongkha.
General Information about the Language
Bhutanese Government Site on Dzongkha
Tashi Delek!
TGF