The short answer is relative majors & minors. If your song starts in C, you can change to A minor, F or G major, or D or E minor. After that, yes you can get away with it, but you need to know what you're doing & how to justify it...

The strongest chord changes (regardless of whether you're changing key or not) are movement by a 4th/5th, followed by movement by a 2nd. If you are going to change key this is where you need to do it, & preferably do it by a perfect cadence (V-I) to emphasize the change...

Otherwise go & study classical theory if you really want it in-depth...