If I need to replace a breaker, can I simply just upgrade it to one with a higher rating, so that it doesn’t trip so often?

Absolutely not! That is a very risky thing to do. Each breaker is sized for the particular circuit it is serving and the wires on that circuit are also sized with a specific application range in mind. If you put a higher rated breaker in and the wire thickness is not also increased, you run a considerable risk of causing a fire as there would be nothing to protect your wires on that circuit from getting hot and possibly burning. And this could end up burning your whole house down.