These numbers may not be entirely accurate, the tests are in early stages at the moment. Look forward to updates.
Leveling
Leveling is the most common way of gaining happiness because it is usually done without even thinking about happiness. If you are leveling a pokemon it will gain happiness pretty quickly. Without any boosting, a pokemon will either gain 5 or 6 happiness per level up. There is no pattern to this and the number is randomly chosen.
Normal = 5-6
Normal + Soothe Bell = 7-9
Luxury Ball = 7-8
Luxury Ball + Soothe Bell = 9-12
Doing Absolutely Nothing
Your pokemon will actually gain happiness without you doing anything. Just stand there with the pokemon out and it will gain happiness every minute you have it out. Without something boosting the gains are pretty slow, you will only gain 1 per minute meaning you would have to wait 220 minutes before you can evolve your eevee into an umbreon/espeon.
Normal = 1
Normal + Soothe Bell = 3
Luxury = 2-3
Luxury + Soothe Bell = 5-6
Walking 256 Steps
Must have your pokemon out. Because of the gains per minute, this one was pretty hard to test but i believe that the gains are static depending on your items as i have not found variations in the amount gained like with the other methods. If you are not leveling your pokemon it would be best to do both this and gains per minute at the same time. The new 50% speed boost from running shoes makes this pretty easy and i would definately recommend having a pair.
Normal = 1
Normal + Soothe Bell = 3
Luxury Ball = 2
Luxury Ball + Soothe Bell = 6
I should also mention that in the pokemon games that the happiness resets on a trade but it doesn't on this