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This “travel at the speed of light” thing. The red shift is used to measure distances right? That means infrared must travel at a different speed than ultraviolet, and that is not getting into the other parts of the electromagnetic spectrum like x-ray or gamma rays. So what does speed of light mean since all the frequencies travel at different speeds? Maybe we could travel at the speed of red light and still not have infinite mass?

The red shift is used to measure distances right? That means infrared must travel at a different speed than ultraviolet

No.

For any wave, speed of propagation is equal to the wavelength times the frequency of the oscillation.

For light, it's

c = f * lambda

Where c = speed of light (constant), f = frequency, and lambda = wavelength.

The frequency for IR is lower than it is for UV. The wavelengths are correspondingly higher. The waves travel at the same speed.

Travel at the speed of light thing?

Good question. The way it works is by that "relativity" phenomenon. You know about how a train coming toward you has a higher pitched horn than the same train going away from you. (Doppler effect) That's what happens to light traveling from a star that has motion relative to the earth. All radiant energy from infrared to gamma travels at the speed of light. If a star is coming toward you at, say, half the speed of light, the light it emits would arrive at 1.5 times the speed of light, right? No, it cannot go any faster. What happens is the frequency shifts upwards. Yellow light becomes blue, for instance. The red shift used to measure distance assumes that the spectrum of a certain type (size) of star is the same no matter where it is. By knowing what size the star is, one could calculate how far away it is and how it is moving relative to earth. Cosmological red shift assumes that the big bang has accelerated everthing outwards and that the farther away it is, the faster it is receding away from us. It happened about 8.5 billion years ago and shows no sign of ever ending. WIKI has a fair explanation. It is theoretically impossible for any object having any mass whatsoever to be moving at the speed of light.