What is the fastest way to achieve the highest level of friendship with a Pokemon?
The fastest way is through lvl up with the soothe bell( not included with the + 5) which gives a + 5 in friendship. Though using the soothing bag in super training will give you a + 20 in friendship though this method is not as fast due to the rarity of the soothing bag.
Have your Pokemon hold a Soothe Bell and walk around with it for a while.
You can also feed it Pomeg, Kelpsy, Qualot, Hondew, Grepa, and Tamato Berries, these will make the Pokemon more friendly, but lower one of their base stats.