As for the moon, the Earth's gravity is what keeps it in its orbit. It's hard to show without vector diagrams. Those can be found in any elementary physics book (or probably somewhere on the web) under "uniform circular motion." The moon's orbit is not circular, of course, but you can get a good idea of what is happening from that model anyway.

Now of course, this does not work at ground level due to air friction (slows the cannonball) and obstructions but get yourself out of the atmosphere and you've got yourself a pretty good orbit. That's what rockets & satellites do. Those satellites (and space shuttle, etc.) are constantly falling to Earth but they are going fast enough (about 5 miles a second I think) to keep missing.

Ok, now you're an asteroid moving through space around the sun (everything is moving around the sun...or around something else that is going around the sun....and the sun is moving around the galactic core... and the galaxy is moving too). You feel the Earth's gravity pull on you but the pull is too weak to totally change your direction. But it does change it a bit.

Remember that gravity is a weak force that decreases rapidly with distance. So the farther away something is, the less speed it needs to escape Earth's gravity and keep going on its merry way.

If nothing moved relative to each other, then yes, gravity would pull the whole solar system into the sun (the biggest collection of mass in the solar system). But with some tangential motion, the planets can keep circling the sun without falling in, the moons can keep circling around the planets, etc. The Moon is moving fast enough for its distance to keep missing the Earth.

If you were to magically & suddenly double the gravity of everything in the solar system, then stuff would start colliding because their current speeds are not sufficient for that gravity at their current distance.

Now, orbits are not perfectly circular....they're somewhat elliptical (oval) to varying degrees (planets can be nearly circular whereas long-period comets have incredibly stretched out elliptical orbits). So, there's a lot of gravitational nudging over time and speeds/directions keep slightly changing. Eventually stuff does crash together but it's much less frequent now than it was during the early formation period of the solar system (because all the stuff in 'poor' orbits has already collided and mainly the 'stable' orbits remain).

interesting side note:
'Stable' orbits are not necessarily permanent. Our good friend Phobos (moon of Mars) has a deteriorating orbit and is due to crash into Mars (or break apart) in about 50 million years. You wouldn't notice a chance in your lifetime (looks stable).