if the laws of physics are the same where-ever you are, there is symmetry in space
if the laws of physics are the same when-ever you are, there is symmetry in time
These two define the Newtonian universe, assuming universal space and time, and give rise to the conservation of momentum and the conservation of energy respectively.
Yet the Newtonian laws operate in a world-model that is very different from everyday intuition. Because Newtonian space is infinite and homogeneous, Earth and its surface are not special places. The directions up,down, and sideways are fundamentally similar. Nor is rest privileged over uniform motion. None of these concepts matches everyday experience.(Wilczeck)
If the laws of physics are the same at whatever your velocity, there is symmetry in spacetime, and the consequence is the special theory of relativity ("I have as much right to claim to be at rest as you do, for constant linear motions").
if the laws of physics are the same, whatever your acceleration, there is new kind of symmetry in spacetime, and the consequence is the general theory of relativity("I have as much right to claim to be at rest as you do, for accelerated motions (I am at rest but in a gravitational field – that's what is causing the force)"
As our description of fundamental processes becomes more complete we see more, and we see differently. The deep structure of the world is quite different from its surface structure. The senses we are born with are not attuned to our most complete and accurate world-models. I invite you to expand your view of reality(Wilczeck).
"Play with some of these, asking what the lived-in world would then be like"
Each time you find the rules for a process unchanged for a kind of change in motion, space or time, there is a symmetry. These symmetries make reasoning about a process much easier, and recognising such symmetries is a way to make sense of the world by recognising patterns.
There is one more you might choose to avoid, at least formally:
If the laws of physics are the same which-ever direction you take as your point of view, there is symmetry in spacetime, and the consequence is the conservation of angular momentum
Avoiding the quantum realm (so spin and supersymmetry) that's all the symmetries associated with space, time and motion—there are more abstract symmetries associated with the patterns remaining identical under other changes. Those might be left for later, or developed only for those students who show particular interest in more abstract reasoning. These symmetries lead to the conservation of charge and other quantum numbers.