If you have Netflix, then I'd recommend using something like Mediahint which can work as a proxy to make the Netflix servers think you're connecting from Denmark.
The Danish Netflix doesn't have tonnes of Danish films, but everything on Netflix is subbed in Danish. There are also a lot of Disney films that are dubbed into Danish and are a great way to practice Danish.
I watched The Road to El Dorado (an old favourite of mine) in English, and then afterwards in Danish (you can change the language freely) and that was a great way to practice.
Let me know if you have any questions about setting it up, but I think it's quite easy.
Native Danish here. I've heard that Danish is quite difficult, especially pronunciation.
You could give Duolingo a try. They just opened for Danish for English speakers lately, and they're almost out of beta stage for Danish!
https://www.duolingo.com/course/da/en/Learn-Danish-Online
I can totally vouch for Duolingo, I used it for Spanish about a year ago. Really good!
It's a paid site (and it looks like they focus more on teaching phrases/conversation rather than the nitty-gritty details). Edit: Just tried the free demo. I think it falls short, especially in the test sections because they only accept one answer (where there may be multiple acceptable ways of phrasing something).
Have you tried Duolingo? It's free and a great starting point.
Yes, mine/dine/sine are used for plural nouns.
Mit/dit/sit are used for neuter gender (-et) nouns, (e.g. huset/mit hus, ansigtet/dit ansigt, æblet/sit æble, etc.)
Min/din/sin are used for common gender (-en) nouns, (e.g. katten/min kat, bogen/din bog, koppen/sin kop, etc.)
As for how to figure out the grammatical gender of a noun: there are no hard and fast rules--it's by and far easier to memorize as you go along/through practice. That being said, there are a few pointers out there (one I can add is that a good chunk of nouns--like ~75-80%--are be common gender), and if you're ever unsure, you can look it up in the dictionary.
Edited for clarity.
Duolingo is probably one of the best (most people love Duolingo, however I learnt Danish before duolingo was popular.
However there are also some good free lessons on SpeakDanish.dk
> For pronunciation: http://www.forvo.com/[1] Most of the words are pronounced "as written", even though that is not how you would say it in every-day speech - we tend to shorten words when we speak. It's still a good resource though.
Another brilliant resource for pronunciation is http://ordnet.dk/ddo, which has both definitions (= betydninger), synonyms (= synonymer), common expressions (= faste udtryk), pronunciation (= udtale) and etymology (= oprindelse).
Complete Danish is a very good start for anyone who is a total, first time beginner. Its broken into 24 chapters, all of a different theme and also comes with audio so you're not just guessing what the words sound like! From a Dane to a learner, take this as a recommendation :)