I truly believe that taking steps, even little by little is the way to see the change. It can be too much to adjust to if it's too dramatic all at once but you are right, by cutting down bit by bit and even going vegetarian first, it all helps with the adjustment to take it in your own time.
Some people that come to me go vegan right away and are successful and that's wonderful but there is usually a powerful reason that drives them.
Other people take it in tiny steps and ease into their comfort zones. No way is wrong, every body's body is different and it's just figuring out the best way for them :)
If you find your why you'll be able to make the changes you want :)