Ok. Then after you marry, you will need to file an adjustment package. This includes the following forms: I-130 (Petition for family member), I-485 (application for Lawful Permanent Residency), I-765 (Application for work permit), I-131 (Application for Advance Parole), G-325a (Biographic data - one for each of you), I-693 (Medical exam that a certified doctor must fill out), and I-864 (Affidavit of support). You will need to file each form with supporting evidence and appropriate filing fees. You can find these forms at www.uscis.gov/forms. The exact same set of forms would be filed for the child if you marry the mother before the child turns 18.
The supporting evidence that you would file would be birth certificates, marriage certificate, divorce certificates if either of you have been married previously, proof of your U.S. Citizenship, proof of your spouse's legal entry into the U.S., and financial documents to prove your income over the last year at least. The documents for the child would be primarily the birth certificate and copy of the passport and proof of legal entry into the U.S.
In about 3 to 5 months after filing, they should get a work permit and Advance Parole. About 5 to 7 months after filing you should get a marriage interview (though now it is taking longer, so maybe closer to 7 to 9 months). If all goes well, a few weeks later they should get a Residency status (green card).
No, they do not have to go home during this process and in fact they cannot go home until they have at least the Advance Parole approved. As far as school and work, they are not supposed to, but they would not be denied Residency if they went to school or worked without authorization.
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