We usually assume the path is piecewise differentiable. For instance, the path might form a square, but along each of the four sides, p(t) is differentiable. Of course infinitely sharp corners are impossible in the real world, so our paths are usually differentiable throughout.
By definition, the velocity of the particle at time t is p′(t). Actually this is the only definition that makes sense. For small intervals of time, the average speed along each of the coordinates is given by a difference quotient. The limit, as time shrinks to 0, is the derivative p′(t), where derivatives are taken per component. This gives the instantaneous speed in each of the orthogonal directions. Together these "speeds" form a vector that gives the overall velocity. For instance, we might be moving 2 meters per second east, one meter per second north, and six meters per second up. These are the derivatives of x(t), y(t), and z(t). The resulting velocity vector points east north and up, but mostly up. It is tangent to the path.
The speed of the particle is the norm, or length, of the velocity vector, which is given by the square root of 12+22+62 = sqrt(41). An exercise in limits shows this is the limit of the particle's average speed as time intervals shrink to 0.
If we are only interested in the direction of travel, divide the velocity vector by speed. This gives a unit vector that indicates the particle's direction of motion. This vector is not defined if the particle is sitting still, with speed 0. Again, the directions of the small line segments, as time intervals shrink to 0, approach the instantaneous direction p′(t)/|p′(t)|.
The change in velocity, or acceleration, is p′′(t). We can develop formulas for the magnitude and direction of acceleration, just as we did with velocity.