Earth and Jupiter gravity assist
This tutorial mainly targets people that are struggling to reach Jupiter and other outermost planets either due to a lack of sufficient launch vehicle or the payload being too heavy (>=2 tonnes). Remember that it is easier to use MechJeb's maneuver planner if you're fully capable of doing direct transfer to a desired body on your own.
First thing you need to do is to download MechJeb. Current installation method of this mod you can find here, for a more in-depth MechJeb tutorial visit PVG Bible.
Once installed and in the game you can find a MechJeb button in top right corner. Click it and select "Ascent Guidance", in it's window change the mode from "Classic Ascent Profile" to "Primer Vector Guidance" (PVG for short). Make sure to activate "Auto-warp" if it's not enabled by default, if you can't find that button click on a button labeled "OPTS". Then select the Moon as a target in the map view, click "Launch into plane of target" and "Engage autopilot" at the top. Don't forget to adjust the orbit values to something reasonable, space in RSS starts at 140 kilometers and you don't want to enter into an "orbit" that is below that number. This allows you to launch your rocket precisely into an ecliptic of the solar system since the Moon is rotating in similar inclination and you're launching into Moon's plane.
Also don't forget to install Precise Node mod or something similar to it. It is really useful for tiny corrections!
Remember that this is not a rocket design guide, so the shown craft is not the point of the tutorial.
First thing you need to understand is planetary alignment. Earth-Jupiter gravity assist is available every Earth year, so our main goal is to figure out when to launch. The window is available when the Earth-Sun-Jupiter angle is about 85-70 degrees. What this means is that if you were to imagine a line coming from Earth to the Sun and then to the Jupiter you would have an angle. Here's the picture for better understanding.
Next step assumes that you're in orbit around Earth already using the MechJeb method described earlier.
The plan is to send the craft on a eccentric interplanetary orbit with the time to Sun Apoapsis of about ~370-375 days, then perform a 100-200 m/s inclination adjustment maneuver and a ~900-1000 m/s retrograde burn to put us on a Earth encounter trajectory that would then slingshot us to Jupiter, after that point you could figure out yourself how to use Jupiter's gravity to travel to Saturn or other gas giants.
Starting with the maneuver in low Earth orbit, it's a ~4300 m/s burn. Remember to adjust the position of your maneuver node in low Earth orbit using Precise Node (the parameter is called UT), this allows you to find the cheapest way to do the burn and obviously if you're ejecting from Earth's sphere of influence in the wrong direction you'll end up in more or less Venus orbit, rather than Mars.
How to figure out whether the burn is correct or not, you may ask. Look for the time to Apoapsis on your planned interplanetary orbit. This time should be somewhere between 370 and 375 days. In my case it's 373 days.
After performing the maneuver next step you should do is leave Earth's sphere of influence (SOI). Unfortunately KSP doesn't allow us to target the Earth while we're still in it's SOI.
Once done - select Earth as a target. You can notice that your inclination relative to Earth is not exactly 0, (about 0.5-1 degree difference), you can fix that if you perform an inclination change maneuver near Ascending or Descending nodes, making them both equal to 0.
After that place a maneuver near your Apoapsis and make it a retrograde burn of about 900 m/s (you can play with that value and figure out what works for you here, I'm giving you an approximation). Don't fall for 100-200 m/s retrograde burn that gives you an Earth encounter! It won't work.
If you can't see your close approach markers it is usually the case when you didn't adjust your inclination, make sure to do that, but here's the tip for how to figure out what's going wrong with your trajectory. Place a maneuver that changes your trajectory in some way, then create another empty maneuver that you manually drag on your orbit. This way you can figure out whether you're coming too early or too late and depending on that adjust your maneuver in low Earth orbit.
Here comes the part where Precise Node mod help a lot. After you've got your Earth encounter make sure to adjust your approach to make it as close as possible to Earth's surface at the same time making it more or less match Jupiter's inclination (i.e. using Earth's gravity to change your inclination relative to Jupiter). Look closely for your orbit past the gravity assist! If you come from the wrong side of the Earth you'll end up pushing yourself into the inner solar system to Venus and Mercury instead of the Jupiter.
At this point you should select Jupiter as a target and fiddle with your maneuver nodes. Unfortunately there is no ultimate solution to how to setup these very precise burns, you'll just have to figure it out yourself for each launch.
As you can see in the screenshot above I'm not getting a Jupiter encounter no matter what I try. This usually means wrong launch time, looking at the screenshot above I would try launching later, you may encounter the same problem since we're essentially eyeballing our launch windows. Depending on your orbit you may need launching later or earlier, on the screenshot above you can see me coming way later which means I should wait for the window a little bit more on a launchpad. If you're noticing that you're passing your Apoapsis sooner than Jupiter arrives there - you should've launched earlier.
Here's the screenshot on which I've got it right. Note the Earth-Sun-Jupiter angle being pretty close to 0 - it's unclear to me why the angle difference on some windows might be so great. The point is - look at the orbit you end up with and adjust depending on the situation i. e. launch earlier/later.
I'm sure you can easily figure out how to use Jupiter's gravity to travel to other planets after that since it's gravity is extremely powerful. Here this tutorial ends.
