Billund is great for families - Legoland, Lego House, Lalandia and WOW Park etc. - but maybe not so interesting for a week for a couple with a one-year-old. Accommodation in Billund is also expensive compared with places a few km away.
But it is a great entry point to Denmark as a short drive in any direction gets you to numerous interesting places. Don't worry about driving in Denmark - roads are excellent and compared with the UK traffic is very light.
North of Billund is Silkeborg, which is the main town in the Danish Lake District. There are activities in Silkeborg (Aqua, the paddle steamer, Tollund Man in the museum) and small lakeside beaches at places like Sejs.
Going North-east takes you to Aarhus, Randers (for the Rainforest) and Djursland, where Ebeltoft is a lovely old village. There's also Djurs Sommerland amusement park, but again, this is for older children and families.
East of Billund - Vejle and Kolding. The Okolariet ecology centre in Vejle is interesting for adults as well as for children. Kolding has a castle (Koldinghus) in the town centre, a big shopping mall (Kolding Centre), a nursing museum and a design museum.
South-east - The island of Fyn, with Odense and all things about Hans Christian Andersen, a railway museum, an open air museum. Or continuing south on Jutland there's Christiansfeld, and right down south near Germany is Sønderborg, a bustling coastal town with a castle. You could also visit Flensborg in Germany.
South-west - Tønder and Ribe. Historical old towns with links to the royal family. Esbjerg, which has a maritime museum and is the southern starting point for the long beaches that stretch the entire length of Jutland's west coast.
West of Billund - The Jutland west coast. Hundreds of km of sandy beaches with occasional coastal towns comprising of summerhouses and camping sites, or based around a harbour (e.g. Hvide Sande and Ringkøbing).
If you want to stay in a city, then perhaps consider Aarhus and explore the eastern side of Jutland. If you want beaches and tranquility, then book a summerhouse or a holiday apartment at somewhere like Henne Strand. Granted, these are small quiet places, but for many that's exactly the attraction - you're away from the hustle and stress of a big city and can enjoy the nature, empty beaches and glorious sunsets.