How is celestial navigation done on a low-level? When we send a probe off to Jupiter or Saturn, or even Earth orbit, how are the rocket firings timed and coordinated?
For instance, when I want to drive to another city I pull onto the highway and use the lines on the road to adjust my steering. I also use intuition to adjust my velocity, even though that won't make a difference where I wind up (only when). However, spacecraft don't have lines on the road to line up with. Nor do they have a direct measurement to take for speed, which is critical in orbits and in planetary transfers.
I suppose that some bright stars near opposition or off the ecliptic (Capella?) and GPS could be used for interplanetary navigation or LEO, but how was it done before GPS? Inertial navigation? How did the Soviets get the Vostok to enter a clean orbit (I understand that the Sputnik orbit was very elliptical)? How did early interplanetary probes orient themselves?
 A: GPS is now used for low Earth orbit, at least by the space shuttle and ISS. GPS satellites are 20,000 km high, so a LEO of 300 km above the surface isn't much different to being on the surface.
Finding the satellite after launch relies on the carrier signal from the onboard communications and a reasonable estimate of where you were aiming the rocket. If the craft is not transmitting you can use active radar - but if it's not transmitting there probably isn't much point in knowing where it is. 
For the craft finding its own position it uses a combination of sun trackers (especially for Earth orbit when you only need an approximate position to point the solar panels) or magnetometers to sense the position in the Earth's magnetic field.
For more accurate pointing, e.g. for space telescopes, star trackers are used. By imaging at least three fixed stars (or planets) you have a very accurate fix on your attitude. Knowing your position accurately is generally less important, since the stars are so far away that where you are (to within a few km) is pretty irrelevant compared to knowing where you are pointing.
The Apollo missions used dead reckoning to fix the position, the speed of the spacecraft was determined from the Doppler shift of an accurate radio frequency sent from Earth and returned by the spacecraft's radio. This together with the time and the orbit gives you the 3D position. They also used sextants viewing the limb of the earth/moon or fixed stars to mostly get the attitude (3D orientation) of the spacecraft so that it was properly pointed for any engine burn.  See How Apollo Flew to the Moon or listen to the author's podcast.
