Q: How long in Earth-time would it take a rocket (average sort of rocket speed assumed) to reach the Moon? Same question for Mars, Jupiter, and the Sun?
The Moon is about 240,000 miles away. To get into Earth orbit, you need to zip along at 17,500 mph. To reach escape velocity from Earth, you must up that to about 23,000 mph. If you traveled at the speed of the Apollo astronauts - who roared atop Saturn V rockets to the Moon in the late 60s through early 70s at 23,000 to 24,000 miles per hour - it would take a little over 3 days (Apollo 11 did it in 3 days, 3 hours, 49 minutes).
But wait - 240/24 = 10 hours, right? What's up with these numbers? Why doesn't a trip to the Moon take just 10 hours? The answer lies in physics.
If you don't care about slowing down to land safely, no biggie. Go ahead and crater into the surface of the Moon 10 hours after launch - it'd be exciting for those of us here on Earth to watch! If you miss, though, you'll keep zooming along past the Moon. If you swing by closely enough, you'll even get a little added gravitational boost to scoot you fast enough to reach the outer Solar System.
This is because escape velocity from the Moon is only about 5,300 mph. Faster than that, you'll just fly by (unless you hit, of course). So, because you need to slow down for insertion into lunar orbit, you lose a lot of time. Then you need to orbit a bit to prep for landing, find a good spot, and so forth.
Now, if you poodle along at highway speeds (just play along - you're using a dark-matter drive so you can ignore escape velocities and such, okay?), it would take about 145 days. Get a co-driver, unless you plan to sleep, in which case add a few months. On the other hand, Apollo 10 holds the record for the the fastest any human has ever traveled: 24,791 mph.
To reach Mars is another matter. At its closest, Mars is 34 million miles away; at its farthest, 249 million miles. Its escape velocity is only 11,000 mph, so you'll want to decelerate a bunch once you get there. But at, say, an average velocity of 25,000 mph, you'll reach Mars in 9960 hours, or 13.6 months.
Flying to the Sun - 93,000,000 miles away (about 400 times farther away than the Moon) - wouldn't take you 400 times as long, because you wouldn't need to slow down at all. That's because the Sun's escape velocity is 1,380,000 mph. That's MILLION miles per hour. Heck, you'll even speed up as you approach this flaming mass of incandescent gas, eliminating even more travel time. Let's say it'll take about 3000 hours (a little over 4 months).
Same deal for travel to Jupiter, which has an escape velocity of 133,300 mph. Basically, just drop as deeply as you can into every planet's gravity well along the way to help get you going as fast as you can, because you'll have trouble attaining speeds higher than its escape velocity without bringing along lots of additional propellant for your fusion drive or whatever. It lies about 391,000,000 miles away at its closest, 577,000,000 miles at its farthest. So, traveling along at 30,000 mph, you'll reach Jupiter in 13,000 to 19,000 hours. Just call it at least two years.
I hope you enjoyed!