It is perfectly feasible to take your bikes on the majority of Danish trains, we have used it several times (in fact it is the only way to get from Fyn to West Zealand if you are intending to cross at Kørsor or Nyborg as there is no ferry anymore and you are not allowed to cycle over the new bridge). We found that most of the information & ticket offices in the stations were quite helpful and most spoke good English.
You will need to purchase a separate ticket for your cycle from the ticket office. Prices are not as cheap as in Britain but reasonable e.g. On the train system around Køpenhagen we went 5 zones ( Solgt Østerport to Skodsborg) and the fare cost us 27.50 DKr per person and the cycles 20 DKr each. If you are going any distance I would recommend booking yourselves and the bikes in advance as there is only a small number of places for bicycles on some trains.
Most of the stations that we used had good access to the platforms for heavily laden touring cycles i.e. ramps & lifts etc. and there are special places reserved for cycles on the trains, these are clearly marked by cycle symbols on the doors.
BUT when it came to getting on the trains !!!! (see picture left).
As you can see from the picture there are a massive 3 steps up to the carriage from the platform!
This is very much hernia inducing and back wrenching stuff so unless you wish to do yourself a mischief think about unhooking those panniers before the train arrives. The DSB recommends that you remove luggage from your cycles to conserve space inside the carriages anyway.
Apparently cycle trailers can be transported as ordinary hand luggage as long as they measure no more than 100 x 60 x 30 cm when they are folded together and the wheels removed.
We understand that it is possible to take tandems on certain trains. The DSB publish a booklet 'Bikes and Trains' and our 2002 copy quotes :
"Usually, you may bring only a regular bicycle on the train, No delivery bikes,tandems or trailers are allowed. However on most trains (except IC3 and S-trains) tandems are allowed subject to to space."
The booklet from the DSB 'Bikes and Trains' in Denmark (in English) is available at most stations. Contact DSB (The Danish State Railways) Tel: +45 7013 1415 for more details and current regulations.
The DSB site has a handy route planner so you can workout routes and train times.