What is it about Relativity? We can observe that there is no special Inertial Reference Frame (IS) that is at rest. To measure a speed you always have to measure it with respect to something else. And no physical laws depends on the speed of its IS, so each observer can regard his own IS as at rest with the same right.
Principle of Relativity: In physics, the principle of Relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of Special Relativity the Maxwell equations have the same form in all inertial frames of reference. In the framework of general Relativity the Maxwell equations or the Einstein field equations have the same form in arbitrary frames of reference. Source: Wikepedia.
Special Relativity was originally based on two postulates: the Principle of Relativity and the constancy of the speed of light.
The first principle is universal but the second is a particular property of light. In this article I explain an attempt to drop the light postulate from Special Relativity as described in 
We can deduce a Lorentz-like transformation with an undetermined invariant speed only based on homogeneity of space and time, isotropy of space and the Principle of Relativity. We don't need to assume the constancy of the speed of light. An invariant speed appears in the deduced transformation automatically, which acts as a general speed limit not only for light.
The invariant speed that appears in the transformation can be finite or infinite. If we set the speed as infinite the transformation degrades to the Galilean transformation. We can not derive from the postulates alone whether this invariant speed is infinite or finite and what the finite speed is. We have to measure this speed. But if an invariant speed exists at all, it is an absolute limit for everything, not only for light.
We can observe that physical laws are the same in all inertial reference frames (Principle of relativity), do not depend on the direction (Isotropy) and are not dependent on the position (Homogeneity). From these 3 observations alone (Principle of Relativity, Isotropy and Homogeneity) we can derive transformations to transform physical observations between inertial reference frames. Lets see how the simplest possible such general transformation looks like (derivation see Relativity without light: A further suggestion; paper by Shan Gao):
general Transformtion between IS
In this transformation, derived from the 3 basic principle of nature, emerges one unknown factor k. k has the units of 1/speed2. So we can replace k with an expression like 1/c2 where c is some speed factor we do not know yet.
Now lets see what this k may be:
k < 0: leads to contradictions in the transformation, so it must be k ≥ 0.
k = 0: if we set k = 0 (which means setting the speed factor c = ∞), the transformation reduces to the Galilean Transformation with Galilean kinematic, no speed limits and absolute space and time.
We know that this is not what is observed in reality. The speed of light is finite and more importantly the same for every observer no matter how fast he is moving with respect to the light source. That means the speed of light is invariant. If it were not invariant, Maxwells discovery of electromagnetism would not be possible and without electromagnetism no atoms would exist, and the whole universe we live in would not exist in this form. So k = 0 is no option either.
k > 0: this corresponds to c > 0 and we get the so called Lorentz Transformation of Special Relativity:
From this transformation follows, that c is a limit for the speed v in any frame of reference, because if you set v > c you get a negative term in √1 − v2/c2 and so an imaginary solution (√−x has no real solution). So c must be a speed limit for v in any reference frame. The speed factor c is not dependent an a specific reference frame or speed v. c is constant and the same in all frames of reference (invariant)!
Note: If the speed v between frames of reference is mutch less than c, then v/c2 and v2/c2 is very, very small and can be neglegted. So if we omit these terms in the Lorentz Transformation we get the Galilean Transformation as a good approximations for low speeds:
So the simple Galilean Transformation may be applied in all situations where the relative speed v between reference frames is mutch slower than this ominously speed c, which we still have to figure out what it is.
The fact that we can observe an apparently static space and time and an almost infinite speed of light is due to the fact, that in our daily life we have only to deal with speeds much slower than c. But all changes with higher speeds.
Ole Rømer was the first to realize that the speed of light is not infinite. James Clerk Maxwell found by experiment two constants, one for the magnetic permeability μ0 and one for the permittivity (dielectric conductivity) ε0, which together gave the propagation velocity c = 1/√μ0·ε0 for electromagnetic waves in vacuum. These constants are independent of any velocity, so they have the same value in each Reference Frame - they are invariant. Maxwell realized that this c is the speed of light measured by Ole Rømer. So that meant that light had to be an electromagnetic wave.
The fact that there exist an invariant speed at all means, that the Lorentz transformation must be the correct transformation, because it contains such an invariant speed constant. Other transformations like the Galilean Transformation contain no invariant values like c or k.
Because we find by experiment (Maxwell) an invariant velocity c for light means, that this c must be the invariant c from k = 1/c2 of the Lorentz transformation. This c must therefore be the highest possible speed for all physical effects, since speeds v > c lead to imaginary results.
From the Lorentz Transformation we can derive a formula for relativistic Energy. The Relativistic Energy formula tells us, that all massless particles have to travel at this speed c. It is also the speed with which causal relationships propagate, since all physical effects can propagate only with a maximum speed of c in a vacuum.
From the Lorentz transformation follow also predictions such as Ttime dilation, Length contraction, Relativity of simultaneity, E = m·c2, the existence of Antimatter etc. All these predictions have been found in innumerable experiments and applications (eg. GPS, particle accelerator, PET scanner) proven to be real and as predicted by Special Relativity.
The speed limit c is not only the limit for the speed of light, but a general limit for the propagation of any cause. Any propagation of a cause greater than c would result in time paradoxes.
Why there is an invariant top speed c in our universe, and not an infinite one, remains an open question. Gao's "guess" is that the maximum invariant velocity could be related to the quantization of space-time at the so-called plank scale. 
With an infinte speed of light there would be no electromagnetism in the known form, and consequently no atoms. We would simply not exist.
General Relativity is an extension of Special Relativity to accelerated frames of reference. It follows logically from Special Relativity although it's math is much more complicated. It makes spectacular predictions as Gravitational lensing, Black holes, Gravitational waves, and much more. Each of these predictions are confirmed by observations. Not only in prinziple, but also in quantity.
Relativity is a thing that we understand very well. It is one of the best tested theories. Only in very special cases it seems to break down and needs some extension. But as much Newton's theory of gravitation is right for speeds much slower than the speed of light and low gravity and is usefull in most practical applications like orbital mechanics, General Relativity is and remains right in the realms of high speeds and strong gravity. Some problems arise on engery densities we can never reach in a lab. We have to use the whole universe as a lab and find evidence for some predictions of proposed extensions to GR to falsify the wrong ones.
At relativistic speeds, different speeds can not simply be added. Otherwise one would be able to get sums higher than the speed of light. If the speed of light were infinite, you could simply add up velocities.
Why is this so? Perhaps we don't measure the right kind of speed?
We can write the Lorentz transformation in a symmetrical form using hyperbolic rotations:
The rapidity has the property that it can take values between −∞ and +∞ and that such velocities can be easily added. If we set v = c, the rapidity becomes ω = ∞. The speed of light in units of rapidity is therefore infinite. The value of 299,792,458 m/s is due to our choice of length and times units, whitch are arbitrary. In Physics they often use so called natural units, where c = 1.