Get yourself a trust set up first of all. It will make purchases and applications easier.
My response will be one based on someone who already has a trust established, but I will give you my understanding of the personal route as well (keep in mind I've never filled out the application as an individual)
It's pretty simple. Decide what can you want , but do your research because there's not really any going back. If it's for a pistol consider the calibers it will be used for, whether you want the ability to service it yourself, or a sealed can, and do research into the manufacturer. Since these are usually a lifetime purchase in my opinion it's best to pay for quality. I like silencerco cans personally.
Next find someone who has it in stock (suppressors can be ordered for you, but it will add a considerable amount of time to your wait since they have to go through a much more lengthy process than your typical firearm)
Fill out your paperwork. A good dealer will walk you through the whole process, and even mail it off for you. Take a check for an additional $200 with you when you fill out paperwork for your tax stamp.
Wait
Wait some more
Google a thousand pictures of your can on your host guns
YouTube
Start to forget about it
When you've finally forgotten about it, your dealer will call you and tell you to come get your new can.
Again, this is all based on applying as a trust. The individual application route adds more time and a few more steps including photographs of yourself, fingerprints, and a signature from law enforcement in your county. Someone else could give you more information on that process. If you want my advice, get a trust. Your life will be easier.
I've done 3 Nfa purchases so far, and built an sbr. The trust saved me 4 sets of fingerprints, and 4 times waiting for LEO signature. If you have any other questions, shoot.