Everything you are asking about is a separate system, and they all take their own tickets, which are not cross-honored. Some details:
The light rail in Edinburgh, for tourists, is only useful for getting to and from the airport (it's VERY useful for this, but doesn't go anywhere else a typical tourist will).
Buses cover the entire city. I used them a lot for short hops, because the city is so hilly. A day pass is sold on the bus for £4 (cash only, no change given). Or, you can buy single tickets for £1.70 (so, 3 or more trips in a day makes a day pass a good deal).
As said above, the central city area is small; depending on where your hotel is and where you are going, you find that you walk everywhere and rarely or ever get a bus. You will need a bus to get to the Royal Yacht Brittannia, which I highly recommend seeing.
Most services in Edinburgh are run by Lothian: https://www.lothianbuses.com/
For trains outside of Edinburgh, these would be bought separately, and would not include local transit in the cities.
For London, you will want an Oyster Card. That is, unless you are part of a travel party with an even number, in which case you will probably want a different kind of transit ticket to get Two For One offers at many attractions. These are 2-4-1 and not half off, so if you are a solo traveler or an odd number of travelers, you can't use them. If you are an even number of travelers, let us know, so we can post that information (needlessly confusing if you can't use it).
The Oyster only seems complicated. I think Rick has the best advice for it: if you're staying 5 days or less, just put on £30, then top it up (add value) as needed. This is called Pay As You Go or PAYG. If you're staying 6 or 7 days, get a 2 zone, 7 day Travelcard on your Oyster, which saves a bit of money.
With the Oyster, you just tap the sensor when getting into the Tube, and tap it again to get out; for buses, you just tap once on entry.
If you are using PAYG, the Oyster computer keeps track, and when you've reached the value of a day ticket for the zones you've used, it stops charging you any more (this is called "daily capping"). So, you will always pay the day ticket price or less - you can't lose. This way, you don't have to worry about peak vs off peak, zones, etc - just tap and go.
If you have a 7 day 2 zone Travelcard, you are covered for unlimited travel in zones 1 and 2 (and unlimited buses anywhere in London). If you want to go outside zones 1 and 2 but still in Greater London (say to Kew Gardens, Hampton Court, or Heathrow Airport), you have to buy the extension BEFORE you get on. When I went from Heathrow to central London, off peak, the supplement cost £1.50 (it's a bit higher during peak times).
Transport for London: https://tfl.gov.uk/
As an aside, have you already bought your flight from Edinburgh to London? If not, the train is faster and easier, end to end (getting to and from the airports, particularly on the London end, adds substantial time and hassle to the flight).