I am a US citizen planning to marry a foreign citizen, divorced with 2 minor children, ages 7 and 10. What would be the immigration status of the children right after the marriage?

They aren't American they are whatever nationality they are now... and your spouse won't be any different either. You have to apply for her and them both. It will take time... and cost money... its not automatic or guaranteed.

My wife wasn't an American Citizen either when I married her. Took just over a year just to get a green card for her (some cases can be far, far more difficult). I recommend starting to find everything you need now before you get married.

Marriage does not in itself affect the spouse's, stepchildren's, or anybody else's immigration status.

After the marriage, you can petition your new spouse and new stepchildren to immigrate (i.e. to get permanent residency). The specifics of how they can apply to immigrate depend on many details about their situation that you have not mentioned here, like whether they are in the US, whether they entered the US legally, etc.

Agreed, your new wife, or her children will not have any status, until you file for them.
Issues of custody of the children, and the new wife ability and legal status to take them out of their area will be an issue also