Housing in Zealand
It is a good choice to locate to a place in Zealand, especially if you are looking for a quiet place, and want to live in a smaller city of approximately 40-50 000 people. There are benefits such as easier access to a place to stay and lower prices as well. Roskilde is cheaper than Copenhagen, and you will approximately be able to save around 25% on rental offers. In case you already have a contract before moving, you could ask the company to help you out to make your life easier on the housing situation. Another typical approach is to ask friends of friends on facebook for help to find a room or apartment.
Rental Price level
The approximate price level of 3 rooms flat of 85m2 in a normal area costs 950 Euro pr. month, according to expatistan.com
Where to look?
Housing in Roskilde or Køge gives you benefits of easier access to a social life
and a connection to the locals and other expats
compared to smaller cities in the region. The transport time to Copenhagen is relative low. You could also look into the cities of Ringsted, Næstved, Slagelse, Korsør, Holbæk, and live a little bit cheaper, but your transport time will be higher, and you will be more annoyed by train infrastructure repairment issues - and you will be more challenged on networking with international people.
Private rent out of rooms/rentals within the average price level
can for instance be found on Boligsurf.dk, Findaroommate.dk or on local groups on facebook. There are other sites and apps for buying and selling like dba.dk, but the site is in Danish. It can be hard to know which site to trust. In general, it is advisable not to believe an offer, which is extremely cheap and looks like a scam.