Officially, you do nothing until the embassy contacts you. No reason to be rushing to file documents that takes a day to do weeks in advance when they haven't asked you to do so yet. To get a good idea how long the wait will be, you should visit forums related to your country. Many countries are different. It's going to take atleast a month or more for them to process it and ship it and for the embassy to receive it for many countries. And on top of that, the embassy has to go over it and then contact you when they are ready for you to start the process.
If anything, the best thing to do is research how to get the police certificate and birth certificate (and divorce, death certificate from past relationship and proof of being able to be married if your country does that) and if there's any special requirements for them. Like in our country, the requirements had to be that the documents couldn't be more than 3 months old. And also, get any documents saying you have vaccines before or they will make you take vaccines you may have already had for the medical exam.
I would add, if the wait seems unusually long, i would contact NVC just to keep in contact. And if they say they sent it, I would contact the embassy too if it's unusually long just to keep on contact.