On my first attempt to get into orbit, I loaded up my rocket with as many engines and boosters as I could strap on and fired the rocket straight up. The rocket breached the atmosphere but to my surprise the rocket came straight back down. I literally thought that launching rockets into orbit meant shooting them really high until they were "in space" haha. I then quickly learned that being in orbit means having a horizontal momentum that is balanced with the gravitational pull of a given body. This balance produces a curved elliptical trajectory.
That's because if you fire your rockets to accelerate you end up in a higher orbit, i.e. your circle is much bigger, and vice versa, to overtake/catch up you want to fall to a lower orbit. That's also not intuitive to someone with "earth-experience".
What's the deal with that?
(1) Your speed is higher; you cover the same distance in less time.
(2) The distance around the earth is larger than it would be if you were lower down.
What's confusing about transfers is that elliptical orbits are orbits where your velocity varies based on where you are in the orbit. Your speed is highest at perigee and lowest at apogee. In fact, your speed at perigee will be higher than the speed of an object in a circular orbit at that altitude. The reverse is true at apogee; your speed will be slower than an object in a circular orbit at the same altitude.
So if you and your docking target are both in circular orbits, and you increase your speed in order to catch up, then you will simultaneously be raising the other side of your orbit, half a turn away. The difference at your current location will be minute, but will grow every continuously over that half a turn. If you're docking in the next few seconds it won't be a problem (the docking mechanism will handle a small offset, and it equalizes your speeds provided the difference isn't too much). If you're still several minutes away, however, you might find that the extra radial velocity means that you fly over the top of your target.
If you miss and continue to follow your new orbit you will indeed find that your overall time to orbit is now higher than your target's. It may be many orbits before you again have a close approach with your target, although if the difference in speeds is small then it may appear from your perspective that you are in fact orbiting your target.
The moon is going much, much faster than satellites in LEO. But it has so much farther a trip, it takes orders of magnitude longer to complete an orbit.
If you're behind a satellite in orbit and want to catch up, counterintuitively you have to decelerate. This slows you, but puts you in a smaller orbit. The effect of the smaller orbit greatly outweighs any loss in absolute velocity, so you catch up to your target.
> I literally thought that launching rockets into orbit meant shooting them really high until they were "in space"
It makes sense that having a high horizontal speed keeps you in orbit if orbit is your goal, but with enough firepower you should be able to make it to a point where you don't fall backwards anymore, right? Or at least where falling is negligible enough that you can stay in that spot for a while.
From the point of view of the game, several of my little green men can attest that coming home from a solar orbit is rather difficult (energy-intensive, time-intensive, or both)
If the plan is to end up in Earth's orbit, you don't want to shoot so far as to leave our sphere of influence. Earth's gravity drawing you towards it is a key component of orbit. But the other component is having enough horizontal (relative to the surface) to keep missing it - and that's what was missing from the parent's "what goes up, must come down" flights.