During the northern summer, the days in Greenland are very long. Always make sure that you get as much sleep as you're used to, as sleep deprivation can lead to all manner of health problems.
During the summer, also watch out for the Nordic mosquitoes. Although they are not dangerous as they do not transmit any diseases, they can be irritating.
Danish and other Scandinavian citizens do not need a visa for Greenland, but your passport needs to be valid for at least three months after your visit.
Generally, if you need a visa for entering Denmark, you also need to apply for a special visa for entering Greenland. Visas for entering the Schengen-area including Denmark do not automatically apply for Greenland; visas are available from the Danish embassy or where you usually would apply for a Danish visa so make sure that you mention that you are going to Greenland. If you stay for more than three months, you need to apply for a residence permit at the police station.
If you stay on the typical tourist paths you do not need any permissions, but any expeditions including any trips to the national park, which by definition are expeditions need a special permit from the Danish polar centre. If travelling with an agency they will usually take care of the paperwork for you. If you are entering or travelling through Thule Air Base, you also need a permission from the Danish department of foreign affairs, since it is a US military area except for children under 15, Danish police and military, US military or US diplomats. See Qaanaaq for details.
The official language - Greenlandic Kalaallisut - is actually that of the more populated western coast. The eastern dialect is slightly different. Both are highly challenging languages to learn, as words are very long and often feature "swallowed" consonants. Try uteqqipugut or Ittoqqortoormiit on for size.
The good news is that almost all Greenlanders are bilingual Danish speakers, and many will even have a functional command of English. Greenlandic words may come in handy for travellers wanting to experience the "real Greenland", though.
Greenlandic is different enough from Inuktitut, the language of the Canadian Inuit who share similar historical roots to the Greenlanders, that the two peoples have difficulty understanding each other. However, attempts are being made to unify the Inuit language, and Greenlandic - with its existing libraries of translated Shakespeare and Pushkin - seems like the most natural option.