Honestly a lot of it comes from experience. Pokemon GO calculates weaknesses slightly different too I think.

Start off by looking up and trying to learn your most powerful pokemons/the ones you use most oftens type advantages/disadvantages. You should try to have a mix of types of pokemon at a decent level, so just learning how their types interact with others will give you a nice stepping stone into beginning to remember stuff.

For rocket battles, your Team Leader will give you a clue about what the rocket trainers pokemon type is weak against if you lose, letting you adjust for the fight, and in raids/gym battles it's directly shown on screen if your attack was weak, strong or otherwise, which you can of course use to help you learn.

Many type advantages have some thread of sense behind them if you think as well, which helps me remember.

Fire beats Grass beats Water is pretty self explanatory for example.

Bug, Ghost and Dark all beat Psychic because they are common fears. (Phobias beat the mind.)

Flying beats bug and plants cuz bird eat them. It beats fighting cuz they have the range advantage. It's weak against thunder and ice because flying in extreme weather is hard. It's weak against rock because of "two birds one stone"

If you can find little ideas for why/how the types work like this it makes them a lot easier to remember I find.