My fiancé is a Russian citizen with a tourist visa that expires October 9. She's coming to New York to stay with me in September and we where planning to get married then. She has a 7 year old daughter back home who's not coming with her in September.
What is a proper way to approach this situation?
We will have to apply for green card right after the wedding, right? What do we do after?
Will she get some sort of document allowing her to leave and come back to the country? Or she would have to stay and wait for the green card? And if so how long does it take approximately?

Thank you for your time

No she needs to get a visa to allow her to come here and get married.
On a tourist visa, it is very possible she will still have to return and re-apply to come back.

You need an immigration attorney

She cannot use a tourist visa to enter the US if she has the intention to marry and file for permanent residence. The correct visa for this is the K-1, and the k dependent visa for her daughter.

After she enters with the K, she can apply for an EAD, and you can marry within 90 days, and then apply for her AOS.

The timing is bad as it is taking about 5-6 months for the USCIS to approve the K-1, which is required for the K visa issuance. The visa issuance is dependent on the US consulate. Good luck.