Soooo sorry for not posting anything for a week! It's been busy around.
I realised that I only saved the post about the DSB Wildcard as a draft. It's short, but here it is:
Go and get the DSB Wildcard on the DSB webpage. You can even get it if you don't have a CPR-number yet (just put 9999 as the last four numbers after your date of birth). The Wildcard costs 185 kroner but if you are going to travel to Jylland or Fyn more than once, you will save money on buying it. After buying it, it's valid for one year and you get discounts on your train tickets! Yay!