It's true that the natural speed limit is the speed of light. So the minimum time required is 1400 yrs. However, it is imposible to reach that speed. We can't even accelerate subatomic particles to that speed.
You asked for the fastest rocket in Earth. The actual record of speed is the Helios II spacecraft (obtained from here) which traveled at 70.22 km/s. Light speed is more or less 300000 km/s. So the speed record is 4272.3 times slower than light, and we would need 5981202 yrs to reach the planet. That are more or less 6 million of years.
Well, of course, we could try to create a very-very fast rocket. Even with 1000 km/s, we need 420000 years. I guess you're not going to become an astronaut.
Time dilation effects are important if you're able to travel at very high speeds. Time dilation means, the faster you travel, the slower the time goes for you. First, even at 1000 km/s time dilation effects are not going to be very important.
And also, you're not going to notice it inside the spaceship. If you go to that planet at 90% of the speed of light and you return, your iOS time app would mark 2800 years, as usual. But my iOS time app on Earth would mark 6042.6 yrs since you left. On the other hand, if you used the 70.92 km/s speed, both time apps would mark 2800 years.
About time dilation you may want to read about the Twin Paradox. It's easy to understand and directly related with our planetary travel =)
EDIT. As @ChrisCundy mentioned in the comments, above calculus has an error -I didn't take in account space contraction. So in fact the time in your clock is less, 1360 yrs.
