Perhaps you know friends who also have searched for a room or apartment before and therefore have some kind of idea of how it is. Ask them about their experience and thoughts on how to go about the process.
No search is however the same, and you can end up with an entire different experience than your friends.
Be realistic and go research
Who doesn't want a big apartment for themself in the center of city for no money at all? Well...Everybody!
Depending on which areas you are searching in, price, size and demand differentiate. Accommodation in Copenhagen is due to the high demand much more expensive than other cities in Denmark. Even though you have set a budget for how much you want to pay in rent beforehand, you might have to reevaluate this if going to some of the more expensive cities.
Copenhagen, Aarhus, Aalborg and Odense are the biggest cities in Denmark and also the cities where most people want to live. Therefore the rent in these cities are higher than other smaller cities in the country.
So before you start your search go check out some listings on vacant rooms or apartments to see what the standard is like in the given city. Then you can prepare yourself and make your search criteria based on your knowledge of what you know about the generel price and size of a room.