How can we tell if a charge is moving in a Magnetic Field? In the equation $\vec F=q\vec v×\vec B$ ($\vec B$ is the magnetic field), what is $\vec v$ with respect to?
If we were to imagine $\vec B$ to be infinitely large. How could we tell if the charge is moving in it or not?
Some tell me it's with respect to magnets. But what happens if take them out and have a parallel and equal field of $\vec B$ in all direction.
If we were moving along the charge in the field $\vec B$ would we observe it having a centripetal force? Because from our point of view the charge is stationary.
Can anyone help me clear this?
 A: The electric and magnetic field strengths and direction are frame-dependent. Therefore the E- and B-fields and the velocity that are used in the Lorentz force expression are all defined in the same frame of reference.
If you wish to consider the force in a different frame of reference then you must transform all of the E-field, B-field and velocity before calculating a new Lorentz force, which will in general be different to that in another frame.
The way to tell whether your charge is moving with respect to a B-field is to look for an acceleration caused by the Lorentz force. This can be distinguished from acceleration due to E-field because the force due to the B-field does no work and the particle speed would not change.
If the particle moves parallel to the B-field then there will be no acceleration.
If you consider the case of a charge moving at right angles to a uniform B-field it will have a circular motion. If you define some sort of instantaneous frame of reference such that the charge is at rest then you will find that there is an E-field in that frame which is accelerating the charge. Note that this electric force will be acting in the same direction as the original Lorentz force but will not be identical and neither will the acceleration (e.g. see Force on a moving charge in 2 frames ).
A: $$F = q v \times B$$
is the Lorentz force in case there is no electric field (otherwise you have to add $q E$). $B$ and $v$ in this equation must, of course, be measured in the same reference frame, which is important because you can change both by transforming between different inertial systems (and on Lorentz transforming a magnetic field you also cause an electric field!).
To tell whether a charge is moving in a homogeneous magnetic field you can look at special examples. A classic example from school or experimental physics lectures is the charge moving with a velocity perpendicular to the magnetic field. Through the formula for the Lorentz force you find an acceleration perpendicular to both the magnetic field and the velocity, the charge will be moving on a circular trajectory.
Another simple example would be the charge moving parallel to the magnetic field: It experiences no force!
A: At the risk of sounding facetious,

In the equation $\vec F=q\vec v×\vec B$ ($\vec B$ is the magnetic field), what is $\vec v$ with respect to?

$\vec v$ is taken with respect to the reference frame of the magnetic field.
In short, magnetic and electric fields are not invariant properties of space. Instead, the fields change if you look at them from a reference frame which is moving with respect to the original reference frame where they were first specified. (The specific transformation laws are here.)
Because of this, whenever we say "this region has a magnetic field", we are implicitly assuming that there is an unambiguous frame of reference and that the magnetic field is observed with respect to that frame. Then, when we calculate the Lorentz force $\vec F=q\vec v×\vec B$, the velocity $\vec v$ is taken with respect to that frame. (And, if there is ambiguity about what frame of reference is in question, then it must be specified when defining the magnetic and electric fields.)
The example you mention at the end is one of the stand-out applications of this principle:

If we were moving along the charge in the field $\vec B$ would we observe it having a centripetal force? Because from our point of view the charge is stationary.

Indeed, from our point of view in the reference frame which is (instantaneously) co-moving with the charge, the charge is stationary, so there is no magnetic force. However, the transformation to a moving frame has transformed the electromagnetic field, and in this frame there is now an electric field, orthogonal to the inter-frame velocity. This new electric field then produces an acceleration of the charge that is identical to that observed in the original frame of reference.
In fact, these symmetries of the electromagnetic field are what inspired Einstein to develop the special theory of relativity, and the paper in which he presented it is correspondingly titled On the electrodynamics of moving bodies. If you want to understand this in more depth, any university-level textbook on electromagnetism will have a chapter on the relationship with special relativity; I would particularly recommend the chapter in Purcell.
A: If you're moving along with the charge, then there is an electric field present.
