I moved from USA to Australia for university, then got permanent residency and later citizenship. I had a pathway because of (1) having completed my tertiary degree in Australia, and (2) having a close relative who was a citizen and could sponsor my application.
Without those things, my only viable option if I wanted to stay after uni would have been to go back to the USA to work for several years in my field of study, then apply for residency on the basis of being skilled in an in-demand industry. Australia maintains a list of in-demand skills, and if your field is on the list then that pathway is open to you - but you would also need to line up a job with an employer willing to sponsor your application ( I think this is mostly just about showing that you will have employment in the relevant field, not a financial burden on the employer, though I could be wrong about that).
My advice would be to make a short list of countries you would consider living in, and then look up their immigration websites and find out which ones you might have a pathway in to (each country is going to be different). If you have family with citizenship in any other countries, begin your search there.