How to prove frame moving uniformly in straight line to an inertial frame is an inertial frame? (Assuming I do not know Galileo's relativity principle and Galileo's transformations and also taking an axiom given in first chapter of landau mechanics that inertial frame is a frame where space is homogenous and isotropic and time is homogenous)
Landau discusses this on pp. 5-6 (3rd edition). First he proves that in an inertial frame A, the velocity of a free particle is constant. Then he points out that in another frame B in uniform motion relative to an inertial frame A, the velocity is again constant. This makes it plausible that B is inertial, but does not constitute a proof. He regards this as requiring additional experimental evidence:
Experiment shows that not only are the laws of free motion the same in the two frames, but the frames are entirely equivalent in all mechanical respects. Thus there is not one but an infinity of inertial frames moving, relative to one another, uniformly in a straight line. In all these frames the properties of space and time are the same, and the laws of mechanics are the same. This constitutes Galileo's relativity principle, one of the most important principles of mechanics.
This kind of discussion is a little vague and fuzzy if you try to fill it all in with perfect rigor, because he hasn't really laid a very complete logical foundation. For example, he casually talks about Cartesian versus non-Cartesian coordinates, but he never defines those terms (presumably because it would require a long digression about other issues, such as the implicit assumption that spacetime is flat, and those issues would be out of place in a discussion at this level). Landau's style is actually the opposite of rigorous, although it's very sophisticated. He tends to make intuitive leaps or argue based on general concepts.
But I think it should be pretty clear that he's right, and this does require experimental input, and can't just be proved given the assumptions he's laid out. For example, it is not true in relativity that we can have a frame B moving at $>c$ relative to frame A. The experimental evidence he refers to is the evidence supporting Galilean relativity, which is in fact an approximation.