Edinburgh public transit

https://www.lothianbuses.com/contactless/ best way is using a contactless card, just tap it on the area beside the driver, it will autamatically give you the best deal as there is a capped amount per day and per week.

That works on the buses. The best way to include the trams is to buy a Day, 3 day, 4 day or 5 day ticket either for City or City and Airport zones on the trams.
Probably the best one for anyone arriving and departing by air is a 5 day city and Airport Zones ticket for £22.00.
Any tram ticket can be bought from the machines or the staff on duty at the airport tram stop.
That is cheaper than buying integrated bus and tram daily tickets at £12 each day, as the weekly bus cap is also £22. So £22 bus cap and £22 tram ticket= £44.

Consider getting a 48-hour Hop on Hop Off ticket. I was there for three days last month and we used the "City Sightseeing" busses which stopped near most of the main tourist attractions, plus were good for general transportation as well. The busses run about every 15 minutes, which allows you to get off and back on with no time constraints for sightseeing. The multi-day ticket allows you to use all three of their colored bus routes (red, blue and green). You can supplement you other travel with single one-time bus tickets, or short distance taxi rides. If you are staying near the city center, you can walk to most of the things you will want to see.

Having just been to Edinburgh last week, we stayed at an airport hotel, so that involved using a different bus company than the city buses. We purchased an all day ticket that included return for 8 pounds each, I believe. Once we were in downtown, we walked. It's rather hilly and was fairly busy with many tourists but was manageable.