(Alternative definition) An inertial frame is a reference frame in which no force is needed to keep a particle at rest.
Intuitively, this means that if all forces acting upon a particle at rest were "switched off", it would continue to stay at rest.
(Assumption) The alternative definition and the original definition of inertial frame are equivalent. In other words, the law of inertia holds in all "alternative" inertial frames.
The below theorems characterize the relationship among inertial frames.
(Theorem) Let K be an inertial frame. Then every point of another inertial frame K' moves at a constant velocity relative to K.
(Proof) Take an arbitrary point P' in K', and place a particle p there that is at rest in K' and upon which no force is acting. Then, due to the law of inertia, p and P' are moving at the same constant velocity in K.
Here we made use of the fact that if there is a force, it must exist in every inertial frame, for it is the "business" of the interacting objects only.
(Assumption) Given two points of K', the distance between their corresponding points in K cannot grow arbitrarily large.
(Theorem) The constant velocity in the previous theorem is the same for every point of K'.
(Proof) Otherwise, the assumption just made would not hold. (I suspect the theorem could be proved without the assumption, but I don't know how.)
In the opposite direction:
(Theorem) If a reference frame K' moves at a constant velocity relative to an inertial frame K, then K' is an inertial frame too.
(Proof) Let p be a particle that is at rest in K'. Then p is moving at a constant velocity relative to K, and due to the law of inertia it can be assumed that no force is acting upon p. Thus, no force is needed to keep p at rest in K'.
(Theorem) The classical theorem characterizing the relationship among inertial frames is also valid in special relativity.