In a circular motion the relationship between the rotational speed (or in other words angular speed) `omega` and linear speed `v` is given by the relation
`v= omega*R` , or equivalent `omega= v/R`
where R is the radius of the circular trajectory.
So the answer is as long as the radius od the trajectory does not change in time the rotational speed increases with the linear speed in the circular motion.
To experience zero weight the condition is that the centrifugal force equals the weight itself. This happens at the top of a circular trajectory if the plane of the trajectory is perpendicular to the earth (see the figure below). Thus
`g = v^2/R`
`v= sqrt(g*R)` , where `R` is the radius of the trajectory.
You need to go upwards (see the figure) to experience zero weight.