Once you are married, you will become a beneficiary of the EU's directive on freedom of movment. The English name for the document you'll need is a residence card. You should not need a visa. (And you won't normally have a "tourist Schengen visa" because you will have entered as a visa-exempt visitor.)
If you look at the page Family Reunification on the website of the Danish Ministry of Foreign Affairs, you will see this:
Please note that special rules apply to EU/EEA citizens and Swiss citizens residing in Denmark under the EU regulations on free movement. Read more under EU and Nordic Citizens.
Following the link takes you to a page that explains the requirements in some depth. It notes that you can apply at the State Administration, but the link there is not currently working. However, a little searching led me to EU residence as a family member to an EU citizen on the "New to Denmark" site, which should be of help.
As far as I can tell, you should probably apply for the residence card within 90 days of entering Denmark, and you should get married within 90 days of entering the Schengen area. If I were you, I would try to get married at least a few weeks before the 90 days are up, and then apply for the residence card on the first working day after the wedding. If you're planning to honeymoon outside the Schengen area, though, you might approach it differently.