When you are married, you can apply for adjustment of status (provided that you are still in the USA). You will need to wait until your green card is approved (or advance parole is approved) before going to your home country. With advance parole, you need to make sure that you won't become inadmissible upon departure. With a J-1 visa and before applying for a green card, you need to make sure that you don't have a 2-year foreign residence requirement.

I offer legal consultations and can assist in matters of the U.S. federal immigration law to clients from all 50 States and internationally. If you’d like to schedule a confidential telephone or email legal consultation, need legal advice or help, please let me know and I’d be glad to help you.

Contact email address: Attorney [@] law-visa-usa.com or LubaSmal [@] yahoo.com . Or you can TEXT a request for a consultation with an Attorney to 1-402-238-1238.

Note: The above response is provided for legal information purposes only and should not be considered a legal advice; it doesn’t create an attorney-client relationship. If you would like to request a follow-up confidential advice on your specific situation and regarding U.S.A. immigration-related issues, we can offer a paid consultation by telephone or email to clients from all States and globally. Please visit our website http://www.law-visa-usa.com/contact_us.html for more details.

It depends on whether you are subject to the J-1 2 year home residency requirement or not.

Assuming you are not, it should be a matter of applying via adjustment of status. Comprehensive information can be found here:

http://www.msclaw.com/Green_Card_for_Spouse.html