1

Best for cycling? Do you mean which regions got the most sights or best roads or...?

Most of Denmark is easy manageable for cycling... It's flat, you can cycle everywhere (except the big highways) and people are use to bikes...

You can put together your own route using the Marguerite Routes (http://www.visitdenmark.com/denmark/attractions/marguerite-route), which is a system of scenic routes with a lot of sights....

2

It has to be Stjernerute Viborg ("star route" at Viborg). They talk about 6 loops and not having to pack your luggage every day.
The English page at visitdenmark does not say the same as the Danish. It offers accommodation packages. The Danish has info on the individual routes. See if you can decipher with google translate.
http://www.visitviborg.dk/danmark/da-dk/menu/turist/aktiviteter/paa-cykel/stjernerute-viborg.htm

To get a star route map, (DKK 75,-)write to VisitViborg (address: Nytorv 9, 8800 Viborg. Tel: +45 87 87 88 88
Email: info(at)visitviborg.dk

Anyway it is great landscape. It sits on the edge of East and West Jutland (I think it is in fact the rim) where the landscape changes from rich soil, beech tree woods and rolling hills to flat, windblown sandy land with fir, pine and heather. At Hald Sø lake you can experience this. A path leads away from the lake from lush green through oak up onto the darker heath.

3

I agree with Ask_G, the whole country is flat and full of dedicated bike trails. The only problem is the wind, which always seems to come from straight ahead. The island of Sjaelland (Zealand) is the most populous area, with the most traffic, but even there you can easily get around on a bike (as the millions of bikes in Copenhagen will quickly show you). Enjoy your rides!