Large reviews of the evidence have not found a consistent, significant weight-gain effect from the combined or progestogen-only pill. Some people notice short-term fluid retention or bloating when starting, which often settles. Perceived weight change can also reflect normal life changes that happen to coincide with starting the pill.
The contraceptive injection is the method most associated with weight gain in studies, and this is a reasonable thing to discuss if weight is a concern for you.
If you start a method and feel it does not suit you, you do not have to stay on it. Side effects, including any weight or mood changes, are worth raising with a clinician, who can help you switch to a different option that fits better.