How fast does a rocket have to travel to get into space?

This really depends on what you mean by "into space." If you just want to get into orbit around the Earth, you need to reach speeds of at least 4.9 miles per second, or about 17,600 miles per hour. If you want to completely escape Earth's gravity and travel to another moon or planet, though, you need to be going even faster - at a speed of at least 7 miles per second or about 25,000 miles per hour.