Most of the museums in Copenhagen are closed on Mondays - go to Roskilde. Visit the medieval cathedral with its royal tombs and wonderful ironwork, then walk down the hill from its west end to the fascinating viking ship museum. It's a group of skeletal wrecks, reassembled after being brought up from where they were sunk to block access, with good interpretive displays, modern full-size rebuilds done by traditional (take a tree trunk and an axe) methods, and, in summer, the chance to take an oar on a short trip out in one of the replicas.
Lots of hands-on activities for children too. It's on the water's edge, with a cafe, and there are more conventional boat trips for those who never fancied being a viking crew member.
Half an hour by train from Copenhagen. There are lots of trains as it is within the local transport net: can use the zone 10 punch strips.
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