Time does not even exist without some frame of reference, as space and time are just ways of assigning coordinates to events and different observers may do so in different ways. And in terms of any particular frame of reference, motion through space is usually thought of as having different positions at different points in time. Since the position varies with the time I suppose this could be described by saying that we “move *with* time”. But the idea of motion *through* time is kind of vacuous since any choice of time coordinate obviously changes when it changes; so I guess we could always be described as moving with time through the same time at “speed” one. This might get more interesting for the case of different reference frames with different time coordinates – in which case one might be moving with the time of one frame through the time of another (with a rate that would correspond to the Lorentz time dilation factor corresponding to the relative speed between the two frames).

So whether you are moving through space or time depends on what coordinate system you are using. In terms of coordinates based on your own body you are always fixed in space and “move” only through time. But in terms of the coordinates of a frame that is moving with respect to you, you are moving through its version of space (with exactly the opposite velocity).