what should I do after getting Q1 visa?

Yes bring your passport and 2x2 pictures and marriage document with your partner to the PSB and convert it to a 1 or 2 year Resident permit. But before going to the PSB you must go to your local police station to register your address or where you'll be living your everyday life and bring that registration sheet to the PSB interview. You "might" be requested to show proof of financial support (Y60,000). So make sure your Chinese partner has that amount of money in their account (not yours) and go get a bank statement to show the PSB. If thy ask you if you work say no and say you guys do stock investment. Dress nicely just like a job interview and try to speak some basic Chinese because the PSB officer in front of you decides how long you stay, so respect goes a long way in China. Next you have to way 10 to 20 days for your passport to be released and after you have it you'll be holding a resident permit that is Multi Entry. But if you and after you get your passport back and you want to move to a new address well than you'll need to go to that districts local police station to re-register your new address because there's a fine if you don't. Police checks pop from time to time to check on registered addresses. I don't know the address, you'll have to google it or find a police station and ask your Chinese partner to ask them for the correct address. Usually only one PSB station can give out visa's. So it shouldn't be hard to find.

BTW, no you don't need a new permit every time you travel, BUT you need to tell your police station that you have returned, just give them your passport and say: I'm back after traveling outside. It's a new policy.