The best way is for them to naturally ask to move out. The fastest way is with amiibo cards. But there are so many glitches and issues right now that "best" doesn't really translate to "safest" so it's hard to say. And apparently now there are issues with voided villagers and more glitched plots, so yeah...
Or sorry, did you ask how to get villagers to move out, as in what techniques to do to accomplish this?
If so, apparently the following has been helpful and fruitful for a lot of people:
1. Talk to at least one villager.
2. Time travel 15 days into the future.
3. Scan your town for a villager with a bubble. If there is none, time travel to the next day and repeat this process until you see a villager with the thought bubble. If it isn't the villager you intended to move out, restart from step 1 after rejecting their move.