As said in the comments, there are only 4 known fundamental forces, gravity, strong and weak interaction, and electromagnetism.
Since we are restricted to low energy classical mechanics, we can ignore two of them, and we are left only with gravity and electromagnetic forces (which includes, among others, magnetostatic and electrostatic forces).
Moreover, in classical mechanics it makes sense to talk about inertial forces, which are apparent forces which appear in non-inertial frames. An example of inertial force is the Coriolis force. In general relativity these inertial forces are described as an effect of the space-time curvature.
In the end, in classical mechanics (non-relativistic, non-quantum), we are left on three kind of forces: gravity, electromagnetism, and inertial forces.
All kind of forces in classical mechanics can be classified in these 3 categories, but in some cases this is not straightforward.
For example: friction, tension, contact forces, capillary forces, pressure, and in general hydrostatic forces originate at a microscopic level from electrostatic forces.
In some cases this classification is a little bit ambiguous. Atmospheric (and oceanic) pressure, for example, is proportional to and caused by the gravitational field of the Earth. However the pressure felt by an object in the atmosphere originates from the many microscopic collisions of atoms of the atmosphere on the object surface. These microscopic collisions are due to the electrostatic repulsion of the electrons in atoms.
Another example is buoyancy, where the net force is a combination of gravity and pressure.
To make a long story short, gravity is gravity, that is, the only force in classical mechanics which originates solely from the Earth gravitational field is the weight of objects. Some hydrostatic forces (like atmospheric and oceanic pressure) are also gravitational in some sense, although one should have in mind that their microscopical description relies on the electrostatic repulsion between atoms.
As a rule of thumb, any time a force in classical mechanics is not clearly gravitational or electromagnetic, there is a microscopic electrostatic origin.