Your spouse needs to maintain the F1/h1 status until s/he get the Ajustment of status approved.

You don't have to be citizen to sponsor and file for your spouse. You can file for Greencard for your spouse while you are on greencard. Only the matter is your spouse needs to maintain his/her status antil the AOS is approved.

If you are fresh greencard holder, it's better to file I-130 immediately so that in 3-4 years your spouse would be able to get approval for AOS.  If you have GC for few years and are about to file for citizenship, you may want to get your citizenship first and file for your spouse after receiving the citizenship. thats will expedite the process.