The network is all managed by TDC, the Danish national provider, so you will probably come into contact with them especially if you need a new line to your home or an old one reactivated. Brace yourself for the price of activating a line to your home and that's even if they don't actually have to visit your house, but once in it all seems to run efficiently.
There are other internet and telecoms providers, and the same as other countries, there are
always deals, one of which might be just right for you. I'm with Cybercity as they do unmetered national calls, and for 10kr a month extra really cheap calls abroad, a handy saving for a foreign Contractor who calls abroad a lot. But you might prefer TDC since they have a good deal going if you take out their cable TV with Broadband, and that seems to offer the best choice of TV channels here. All comes down to what you want.
For phoning abroad more cheaply you can also buy discount phone cards at places like the post office.