Hmmm.... well it's certainly true that DSB's trains are a bit worn out a present and/or have a great deal of grafitti on them. The company is, shall we say, a little starved of cash at present and trains are often a little late, but overall it's a pretty good system with high frequencies on most routes.
The main rail interest (apart from the Railway Museum in Odense, which is recommended) are the small, light railways at the North of Zealand, running from Hillerod to Gilleleje and Helsingor plus a short branch to/from Naerum on the S-Tog network. The 'bus-train' referred to is normally a DMU marketed as 'Lille Nord'. These lines date from the turn of the 19th C and were originally private lines, now operated by a public company known as 'Lokaltog' (although the Lille Nord is DSB's). They are very light in construction and are a bit like the Vicinaux network in Belgium (when it existed), the large forest north of Hillerod (the Gribskov) is particular interesting for looking at the network of lines through it).
If you do go to Helsingor, the frequent ferry connection to Helsingborg is worth looking at to marvel at the operational skill with which four large boats an hour maneouvre into the their berths to maintain a regular connection, almost to the second, between Sweden and Denmark. In the good old days, these used to be train ferries and, if you look carefully, you could see a few clues from this period outside the station at Helsingor.