I was just reading an article on GQ about Justin Beiber (sorry) growing out his buzzcut. Regardless of how you feel about him, there were some good tips in it for not just growing out buzzcuts, but ones men can apply to any type of haircut they're trying to grow out. Instead of repeating what they said verbatim, I'll link their article at the bottom and offer more tips in addition to what they wrote here.
As they touched on in the article, one of the main issues with growing out the buzzcut is how round it gets. This is a big reason I don't like buzzcuts on most men. Since the clippers are following the shape of your head, the haircut ends up really round. As your hair grows out, it gets even rounder. When transitioning from a buzzcut to something new, you'll need to come in to the salon often to get the sides of your hair cut. If you don't quite need that much hair cut, I offer a service called a hairline and neck trim. What I do for this service is clean up your sideburns, tidy up the hair around the hairline, trim the hair on the neck, and do a taper in the back. I do not touch the hair on the sides or the top or give a shampoo with this service. Since it doesn't take me a lot of time I just charge $10. It's good for growing out any type of haircut, not just a buzzcut. It's also a quick way to keep your hair looking it's best as it grows out.
When growing out any haircut it's also a good idea to get it all cut once in a while. If you just let your hair grow without trimming it, you can get split ends which can cause breakage further down the shaft, hindering the process of getting your hair longer. One strategy I have been using over the past year as I've been growing my hair out is using a leave in conditioner. This is a conditioner made to be used outside of the shower and worn in your hair throughout the day. They don't have any hold, so if you need your hair to stay put, you might want to use a hairspray alongside it.
A few other quick tips for growing out your hair--wash your hair as little as you can get away with, be gentle with your hair when it's wet (this is when hair is its most fragile and can break), and try to avoid putting heat on your hair. I know most men don't have to worry as much about the last tip, but the things to avoid are blow dryers, flat irons, curling irons, and any other hot tool. If your hair is long enough to pull back in a pony tail or man bun, try to wait until the hair is dry before putting a band around it to help avoid breakage while the hair is wet and fragile.
Here is the GQ article I referenced earlier in the post.