The major banks here tend to add fees on for everything, and you'll find you might actually end up missing the UK Banks a bit.
Dankort is the basic card you get with an account but it's worth remembering that this card is useless outside of Denmark. A lot of the banks charge up to 500 krone for a VISA/Mastercard, so again it is worth shopping around.
New UK Entrant Northern Rock, as
well as some cheap internet banks such as SkandiaBank and BasisBank are starting to shake things up and banks are having to increase rates to attract savers, so my advice would be to open a basic current account with a High St. bank, then once you have internet and telephone established, open cheaper accounts with other providers. You can still keep your High St. account for things such as withdrawing money over the counter.
For a personal account (lønkonto),
I use DanskeBank and SkandiaBank. Skandiabank pays
reasonable interest on credit banking, all their normal facilities are free and you can use their cashcard at any machine free of charge, which is unusual in Denmark - use your High St. bank's Dankort at another banks cash machine and you will be charged at least 5 krone a time for the privilege. BasisBank's offering seems comparable too and they do offer a free VISA card so could be worth investigating.