Why don’t we consider $g$ for a body along one dimension?

I assume you're talking about the formula: $$ x(t) = x_0 + vt+frac{1}{2}at^2 $$ Where $x_0$ is the initial displacement, $v$ is the velocity and $a$ the net acceleration.

This formula does actually include gravity. Specifically, $a$ is the net acceleration, coming from all forces (gravity frequently being one of them). However, if you are moving on a flat surface then the force of gravity will usually be cancelled by a normal force. You should draw a free-body diagram with all forces included to determine what the value and direction of $a$ should be.

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Let me give two examples:

If you have a body in free fall (e.g. you throw a rock from a cliff), then the only force (neglecting air resistance) is gravity, $F=-mg$. In this case, $a=-g$ and gravity is what gives you the acceleration.

If, on the other hand, you are considering e.g. a car accelerating along a straight road, then the forces acting are gravity, the normal force perpendicular to the car, and the force from the road on the tires due to the acceleration of the car. The normal force compensates the force of gravity, so the net acceleration $a$ is exclusively along the direction of the road.