Disclaimer! Please, double-check the information at Migrationsverket.se!!!!
The first and most obvious one is to find a job. You can apply for it during your education and get a working permit. After 4 years of working, you can apply for a permanent residence permit. There are nuances with this type of visa, you can’t change your position and employer for a certain time.
As a researcher, you can get into fast track and get a special type of residence permit. This one allows you to wait for it in Sweden and start working while waiting.
Looking for job
If you have enough money and plan to look for a job you can apply for a 1-year residence permit and stay in Sweden. Keep in mind that you can’t leave Sweden while waiting for the decision!
Moving to someone in Sweden
If you’ve met the love of your life here you can apply for a permit to stay with your partner. Make sure to apply before your current residence permit expires, so you can wait for the decision inside Sweden! And check the requirements for your partner’s income and living conditions!
Good luck with your decision!