Copenhagen seems to have a very confusing system of payments for travel within local buses and trains (within the city), beyond single use tickets that is. Is there a catch-all travel card similar to Oyster in London that works on both trains/metro/subway and buses?
All local tickets work on both trains/metro and buses.
There is an Oyster-like contactless card called Rejsekort, which gives fares about half of those of discrete tickets. Unfortunately it's not terribly convenient for visitors. The transport agencies are pushing hard to get everyone to use name-registered cards (which require ordering in advance and having the card mailed to a Danish postal address). There are anonymous cards available, but they are quite expensive (a 80 DKK issuing fee plus a minimum balance of 70 DKK) and are only sold from selected ticket offices.
There are vending machines that issue anonymous cards at the airport, at the central station, and at all metro stations.
Leftover balances (including the 70 DKK minimum balance) can be refunded at very few places. Within Copenhagen the only opportunities seem to be the airport, at the main ticket counter at the central station, and at Valby station.
To use the Rejsekort, touch in and out at the card readers in buses or at platforms, recognizable by a glowing blue disk. The card readers marked "check ind" is for touching in, and those marked "check ud" are for touching out. Touching out is optional if you're continuing directly to a different train/bus where you touch in again, but if you leave the system without touching out, a penalty fare will be deducted.
If you're used to Oyster, note in particular that you must touch out when leaving a bus at the end of your journey!
For short visits, two types of 24-hour tickets exist, and can be bought from vending machines.
City Pass (90 DKK for 24 hours) is valid in fare zones 1 through 4, which means a distance of roughly 5-6 kilometers from the city center plus the entire island of Amager (so you can get to the airport). This also exists in a 72-hour variant for 200 DKK.
A plain 24-hour ticket (130 DKK) is valid in the entire Capital Region plus the old Roskilde County -- that is, up to distances of about 50 kilometers from Copenhagen (except Sweden) or everything to the north and east of the cities of Roskilde and Køge.
Then there are 7-day "FlexCard" period tickets which can be good value if you travel a lot during a one-week visit. The downside is that you need to know in advance exactly which fare zones you need it to be valid, and figuring that out is probably not worth it for most visitors-for-pleasure.
2One other option is the Copenhagen Card, a pass that includes pretty much all of the museums and attractions plus unlimited public transit in the whole area during its validity. As with all such schemes, you'll have to do the math to see if it makes financial sense, but it's convenient to use the transit system without having to think about fares. You can also visit attractions more than once, which was a nice feature for Tivoli. May 27, 2016 at 16:03
1@ZachLipton: That's rather expensive too, though -- if it's worth it, it would be due to the specific attractions one plans to visit. I don't think it is physically possible to break even on the 24-hour or 48-hour Copenhagen Card on transit cost alone, relative to a use-and-discard anonymous Rejsekort. The 72 and 120 hour ones might, if you spend more than 12 hours a day on buses or trains ... May 27, 2016 at 17:02
1Oh I'm sure it's not worth it just as a transit pass. 24 hour Greater Copenhagen transit passes are cheaper than that. But it might be worth it if you visit enough of the attractions. Just another option; certainly not always the best one. May 27, 2016 at 17:17