Its about 300 km to Sweden, so its reached in 3 hours on the motorway :-) But It'll set you back around 50£ for the 2 bridges.

It depends on what your interests are but for driving, there are the The Marguerite Routes that are scenic routes through the whole country , marked with a sign with a Marguerite/Daisy. A good start would be along the westcoast to Skagen visiting f.ex. the Bunker Museum in a former German Cannon site with stuff from that period, there is a Church Den Tilsandede kirke, south of Skagen thats half buried in windswept sand and close by is Råbjerg Mile, a inland "walking" sanddune, that is moving 18 meters a year across the country.
On Skagens Gren, the top of Denmark you see the Kattegat and Skagerrak meet, with waves coming from two sides.
Ålborg on the east coast of Jylland has a maritime and an airforce museum and a lot of nightlife. At Moesgård and Silkeborg, you can see the Grauballe mand and the Tollund mand , two well preserved bog mummies. You could sail from Frederikshavn or Grenå to Sweden and return via the bridges making it a round trip.
The Viking ships museum in Roskilde together with The Cathedral, where Danish kings and queens have been buried since the 1500 century is worth a visit.
If you go down the eastcoast of Jylland instead, visit Jelling, with it s two enormous burial hills and famous runic stones, set over the Viking king Gorm den Gamle(Gorm the old) by his son Harald Blåtand(yes THE Harald Bluetooth, that the wireless transmission was named after) in 960.
On Fyn, you could just drive around the Island on small coastal roads and visit the 8 smaller cities situated along the coast.
In the middle is Egeskov Slot, a water castle from around 1400 that still has its moat around the castle. Its built on oak logs, rammed into the old lakebed. On the premises are a Farm museum with a lot of tools and 15-20 horse carts, a technical museum with cars and planes and a large museum for motorbikes and mopeds and a large park around it all.
In København visit the National Museum that has the worlds greatest amount of inuit/eskimo/ greenlandic items in the world. Around 100.000 because around 60.000 were sent to the new Museum in Nuuk in Greenland. They also have a fantastic exhibition about Denmark's history and prehistory with many iron age and viking age finds. North of Kbh is of course Kronborg, Hamlets home.
From Kbh you could go south, see Møns Klint, a high chalk cliff, and take the Rødby-Puttgarten ferry to Germany and make a roundtrip back to Tønder and Ribe(Denmarks oldest town from around 700) and back to Esbjerg.
Its off season so check opening hours.
Most museums are closed on mondays,

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Poul
May you enjoy peace and good health !