Hi stardustshadow and welcome to MR!
As EowynCarter said, you'll have to keep in mind what format ebooks you'll be reading (and lending from the library) are in and - as important - what kind of DRM the library is using.
As far as I can see
over here (although born some kilometres south of Sønderjylland (in Holstein ;o) ) I can't speak dansk but guess some of the meaning) Københavns Universitets Bibliotek has some Sony eReader Hardware on loan too.
I'm not really into this but
any reader you want to use with this ebooks will need support for Adobe DRM as provided with Adobes Digital Editions Desktop Program. You will find more info about this in the Sony-Subforums and the other device Subforums here on MR and in the wiki.
If you want to read pdf-documents with your future reader you'll have to bear in mind too that not all (small screen) devices provide good pdf readability. There is endless advice on this here on MR too.
For myself I did settle on the enTourage eDGe for business-pdf-reading (and annotation) in the end. But that comes with quite some price tag.
Maybe one of the upcoming
new pocketbook devices are of some interest to you.
And, last but not least: For advice with the local market in Denmark ask some of your fellow countrypeople like e.g.
Ea. There are quite a number of them here on MR.
All the best from the Baltic Sea!
beachwanderer