Somewhere around 13,000 mph I thonk
An approximation: it went 350,000,000 miles in about 8 months or 240 days, and 24 hrs in an earth day, so it must have been traveling at about 64,000 miles/hr.
The speed the wave is traveling through space
It took the Curiosity Lander 253 days to get to Mars and the distance between Earth and Mars at their closest point is about 33,926,867 miles. Do the math correctly and the speed at which Curiosity was traveling was about 5,587 mph.
About 8,750 miles.
Depends on the medium through which it is traveling.
Really fast
It would depend how fast you were traveling, but with a space shuttle it would take about a week
When you travel through your mind, expect humiliations, fast speed, and free association.
You need to go fast to break through the atmosphere because the air molecules are denser near the Earth's surface. Going fast provides the necessary momentum to overcome the resistance of the atmosphere and break free into space.
The best fictional way to travel through space fast without messing with time is teleportation.
Slightly faster than the reactions of a cat, since "curiosity killed the cat".
Astronauts can perform their space walk and move around as if nothing is happening while the space station is traveling at 17,500 miles per hour, because there is no gravity. This keeps them from feeling the fast speed.