What's the fastest/easiest way to get Forgotten Souls?

There are several ways for getting Forgotten Souls :

• Doing rifts : The rift guardian has a high chance of droping a Forgotten Soul. Pick the higher Torment level in which you can complete the rift quickly (2-3 min max). Farming in party will help you farm faster and in higher Torment resulting in more Forgotten souls. Moreover, the bloodshards you get from the Rift Guardian allow you to gamble at Kadala, significantly increasing your legendary "drop" rate.
• Killing an Odious Collector
• Salvaging unwanted legendaries (my trick to get "trash legendaries" is to really wish for something good, then the next 50 legendaries are going to be worthless). If you really need Forgotten Souls and have plenty of other crafting materials, you can use Kanai's Cube to upgrade rares into a legendaries and salvage them.

I usually clear T10 rifts and salvage every legendaries that is not usefull for my character. I currently have more than 8000 unused Forgotten Souls.

I've found that doing rifts with a good party is the fastest way. The rift guardian has a pretty high chance of dropping a soul and legendary drops are boosted while in rifts. There is no other good way really of getting souls other than rifts.

As of the latest patch (2.1) you can no longer do the following as everyone needs to contribute a key to enter the rift:

An even faster way now is to join the Rift it Forward community (assuming you have enough keystones to open rifts) to quickly farm Rift Guardians as well as to gain blood shards for gambling purposes.

With this community, you can gain hundreds of souls very quickly.