You could hang around http://www.jubii.dk/ for a while. (This works for us Swedes too.)
The site recommended by Per H. Jørgensen looks really simple. (http://www.danskherognu.dk)
We Swedes might also want to take a peek at this side in order to avoid certain embarrassing traps while trying to make ourselves understood: http://www.danskaspraket.se/fallor.php
I got started with the Complete Danish course from the Teach Yourself series (author: Bente Ellsworth https://amzn.com/1444194984 ). I've also briefly reviewed a similar course called Colloquial Danish which seems to be on the same par. Either course is definitely enough to get you started.
Beyond that, I would recommend looking for books written for children about 12 years or so. Books in this range tend to be grammatically simple while using a great deal of vernacular, conversational vocabulary. Shipping will be expensive (like, $40 USD per paperback book), so if you have a tablet or phone to read e-books on, this is probably the best way to go. And even better if an audiobook version is available to purchase and follow along with. An example you might look up is Djævlens Lærling by Kenneth Bøgh Andersen. In any case, https://www.saxo.com/dk/ is a great place to get started. They sell books in all formats and won't give you trouble trying to place an order with an American credit card.
If you want a more granular, very academic examination of Danish grammar, check out Danish: A Comprehensive Grammar from Routledge (http://a.co/6jcNoV3), though this is more of a reference manual and not a course.
This is an excellent resource for beginners. It includes sound clips, which is very important, as the pronunciation is the biggest challenge when learning Danish :-)
The App "Learn Danish with Babbel" is also good. It also includes sound.