When it comes to space travel, the speed you can reach really depends on what you mean by space. If you're just looking to get into orbit around the Earth, you'll need to reach speeds of at least 4.9 miles per second or around 17,600 miles per hour. The closest star to Earth is Proxima Centauri, which is about 4.25 light-years away, or about 25 trillion miles (40 trillion km). The fastest spacecraft in history, the Parker Solar Probe, is currently in space and will reach a maximum speed of 450,000 miles per hour.
That's fast enough to get from Los Angeles to New York City in just 20 seconds! But it would take the Solar Probe 6,633 years to reach the nearest neighboring solar system to Earth. To travel interstellar distances in a human lifetime would require speeds far beyond what current space travel methods can provide. Even with a perfectly efficient drive system, the kinetic energy needed for those speeds is enormous by today's standards. In addition, collisions with cosmic dust and gas at such speeds would be very dangerous for passengers and the spacecraft itself.
In theory, this could be achieved by creating a “warp bubble” that moves faster than the speed of light. For example, a spacecraft could travel to a star 32 light-years away by accelerating at a constant speed of 1.03 g (i). This approach does not violate the laws of relativity since you don't move faster than light in the space around you. Regardless of how it's done, a propulsion system that can provide continuous acceleration from departure to arrival would be the fastest method of travel. Imagine if you could compress the space between you and your destination so that it's only one meter away! Interstellar travel is much more difficult than interplanetary space flights due to the vast distances involved.
The possibility of making such a trip only to spend the rest of your life inside a sealed habitat and venture out in a space suit eliminates many potential targets from consideration. Scientists are exploring other ways to go fast, including warp travel popularized by Star Trek. The crew of an interstellar ship would face several major dangers, including psychological effects from prolonged isolation, physiological effects from extreme acceleration (if it exceeds 1 g), exposure to ionizing radiation from space and possibly from the ship's engines, and physiological effects from weightlessness on muscles, joints, bones, immune system and eyes. In this scheme, a 30-kilometer secondary sail is deployed at the rear of the spacecraft while the large main sail continues moving forward on its own. The spacecraft itself uses a 12 million ton ball of frozen deuterium to power between 12 and 24 thermonuclear impulse propulsion units. At several hundred million kilometers per hour, every speck in space becomes like a high-powered bullet that hits the hull of a ship. Of course, micrometeoroids aren't the only obstacle to future space missions where higher human travel speeds may come into play.
Speculative dangers could also arise if humans manage to travel faster than light either by taking advantage of gaps in known physics or through paradigm-breaking discoveries. However, shortening travel times would mitigate these problems making faster approaches highly desirable. All rockets - even those used by SpaceX and Blue Origins - burn rocket fuel that's not much different from car gasoline.