In general, Denmark is defined as a quite liberal country and has for decades been considered a front runner in regards to gay rights.
It is illegal to discriminate people based on sexual orientation in workplaces and lesbian couples are allowed state sponsored artificial insemination. Denmark was also the first country in the world to allow registered partnerships and from mid-2012 homosexual couples can get married in the church.
So in terms of acceptance of homosexuals we’re doing ok. However, as in many other countries the acceptance level of gays and lesbians differs depending on what part of the country you visit. Whereas you will probably be just fine walking hand in hand with your special someone or one-night-stand in Copenhagen, you might turn a few heads and provoke some people if doing the same in a small village in Jutland. Be aware of your surroundings and you will be just fine!
If you cannot keep your hands of your partner and just need to kiss him or her in public, consider doing it in a gay café, bar or park where you have some “privacy”. There are plenty of places in both Copenhagen and Aarhus that you can check out for more intimate times.
Please be aware that no place is perfect and hate crimes against homosexuals do happen in Denmark as in other places. In case you experience anything discriminating and criminal, please contact the police immediately at 112 and report it.