This is complicated.

The reason we use g=9.81m/s^2 on earth is because for all practical purposes it's good enough. In most practical purposes you can even use g=10 and be fine.

"Gravity" is the sum of all quantities of mass interacting with each other based on their masses and the distance between those masses.

You can also have perceived gravity, where the rotation of a body such as the earth has centripetal forces that counter some gravity. The gravity didn't change compared to if the earth were stationary but it "feels" lower to some degree because of this.

No planet produces a set amount of gravity, it all depends on the mass of the "test object" (yourself) that's you're using to observe the gravity, and the relative distance between the bits of that planets mass and the bits of your mass.

Likewise the gravity acting up on you because of that planet does not necessarily change when a moon is introduced, instead the moon has a separate effect on you, and may effect the planet in a way that effects you.

Technically a car driving down the street has some gravitational effect on you, but it's so small that it's not worth considering. The interaction between you and the planet you stand on is so much stronger than any other heavenly bodies you see in the sky that unless you're planning on orbiting the planet, it's not worth considering.

If you want to know what you would feel on another planet, first decide on what degree of accuracy/precision you care about. If it's not very accurate, just estimate an average planet radius and the over all mass and use the law of gravitation.

If you need it to be very very very very accurate and preciseyou'll need to figure out the planets precise center of mass, the exact radius between your center of mass and the planets at the point you care about, then do the same for every other massive object nearby until the forces are small enough to be outside your desired degree of accuracy. This should still happen pretty quickly.