Also known as SMK, the National Gallery is the country's highest repository for artwork, and its wide umbrella draws in everything from the Middle Ages to today. If you're going chronologically, start upstairs, where the entire floor is given over to European work, mostly paintings—the Dutch and their light, the French and their fruit, the Danes and their old women.
Although much of the collection outside of the Danish stuff is of the lesser variety (you will find some Rubens, Titians, and Rembrandts), it's still worthy and it's universally well-conserved. Particularly striking are The Judgement of Solomon, depicting a soldier holding a baby by its ankle and on the cusp of cleaving him in two with a sword; and Joakim Skovaard's skin-crawling Christ in the Realm of Dead, depicting Jesus more or less ministering to a horde of zombies. Those are the big works, but there are scads of appealing portraits and smaller works that charm in finer details like brushstroke and expression.
The entire back half of the museum, in a huge modern glass pavilion overlooking a pond in a pocket park (the window-facing amphitheatre is a fine place to rest your feet), is for contemporary art, and here is where the collection loses the thread. Descriptions are overworked and the pieces perhaps too obtuse for most casual visitors, although "Please, keep quiet!", a walk-in mockup hospital ward containing four beds populated by mannequins, is something to which most people can relate.
The SMK is not the friendliest museum you'll visit. All bags must be checked in the sweltering basement lockers, which are in disrepair, and before you lose a coin trying to make one function, first look for the bin of washers first and save yourself the DKK 10.