The key lesson you need to remember about orbit (in KSP or real life) is that orbit is sideways speed, not up distance or up speed. Going up, you will come back down. Go sideways fast enough, you'll come down but keep missing the planet below you, so you stay in space.

The most efficient way to do this would be to just go sideways really fast from the ground. Problem: there's a lot of air in the way. So the right thing to do (again, both in Kerbal and in real life) is to initially go just up, but as the air reduces, gradually switch to moving sideways instead.

Then once you're up there, you need to know the basics of manipulating orbits. Lesson number 1: you can increase the height of your orbit on the opposite side of the planet by increasing your speed in the direction you're currently going. So once you get your apoapsis (highest point) above 100km or so, you can turn off your engines, wait until you get there, then burn hard in the direction you're going and your periapsis (the lowest point of the orbit) will raise up like magic.

You'll be orbiting in no time.