We were taught in school that the law of inertia indicates that an object tend to stay the way it is, so if you throw something in space it will tend to go on forever and ever. The reason an object falls down when you throw it on Earth is because of gravity and air resistance. If that's the case, why don't rockets and spaceships need just enough fuel to escape the atmosphere plus the single thrust to push the craft in the right direction and let inertia drift it away?
That's exactly the case. If you look at the trajectory of any given spacecraft, you will see that it has a few burns of the rocket engines punctuating very long periods just coasting along in orbit around some other body.
For example, the flight path of Apollo 8 has something like eight different rocket burns: launch, translunar and transearth injection (to get out of orbit and go towards the other body), three course correction burns, lunar orbit insertion to catch up with the moon, and one orbit correction burn on the moon.
Image source: Wikipedia
The rocket engines spend most of their time turned off, and carry just enough fuel for all of this plus a little extra for safety. This still means that the initial rocket needs to be huge, because the translunar injection requires quite a bit of fuel and that fuel needs a huge other load of fuel to get into orbit.
I suspect that you may be under the mistaken impression that there is no gravity in space. This is a common belief since we all can see the astronauts floating in "zero g" when on the ISS are some other spacecraft.
However, we can easily dispense with this misconception by asking "what keeps the ISS in orbit around the Earth if there is no gravity?". Of course there is gravity; Earth's gravity keeps the ISS circling 'round the Earth.
The reason the astronauts float and experience "zero g" is because they are in free-fall.
In fact, when the Apollo astronauts left Earth orbit at about 25,000 mph, their spacecraft was slowing down and, by the time the spacecraft begin falling towards the Moon rather than the Earth, their speed had fallen to, if I recall, around 2000 mph (I'll verify and update later if necessary).
So, you see, it isn't the case that, once outside the atmosphere, a spacecraft will maintain its speed, relative to the Earth, without firing the rocket engine. That would only be the case if it were in space far away from any other gravitating body.
The escape velocity from earth (the speed required for an object to leave earth completely, i.e. travel infinitely far away) is 11.2 km/s. If the object has a smaller velocity it will return eventually.
Unless an object is launched straight up, it also has a sideways velocity. This means that it will not fall back directly on top of the launcher. If it moves horizontally at a speed of 7 km/s or more, it will keep "falling" beyond the edge of the earth and stay in orbit. Of course, you need to do this high enough so that you are out of the atmosphere.
Spacecraft initially travel vertically to escape the atmosphere. When they are high enough they start changing direction so they eventually travel about 7 km/s "horizontally". At this point they can stay in orbit indefinitely. To travel to other planets, a further boost changes their orbit so that it becomes large enough to eventually reach the other planet. En route it may get more boosts, either from on-board rockets, or from other planets.
This is a wonderful question and filled with interesting albeit incomplete or inaccurate answers.
Launching an "orbital body" concerns two issues and two issues ONLY namely mass and intertia.
So before we "launch" anything we must first understand THE EARTH IS MASSIVE...which works AGAINST the "inertial reality" of a conical shaped item which albeit is very MASSIVE (weight, size, volume, etc) is still "just sifting there."
Does one need to compute the Earth's Mass? No, not at all. Simply the knowledge that the Earth is truly massive AND IS AN ORBITAL ITSELF is sufficient to become an expert in orbital mechanics.
In other words you need no understanding of physics AT ALL to launch a rocket.
What you do need to know is the Earth wants to "throw" this object so to paraphrase a famous term "the more massive the better."
What "the physicists" get wrong is that getting a rocket off the ground has nothing to do with physics at all but merely "dumping the mass of fuel on the ground." Since the mass of the fuel is far larger than the mass of the object...the object is physically "lifted off the ground."
Everything that comes after said "lift" is just a function of performance and working against the natural forces having lifted the object simply wanting to fall.
The United States struggled MIGHTILY with this problem in the 1950's I might add...something the German Scientists solved quite easily.
This involved "controlling the dump" (to dramatic effect with the Saturn V) as it were...but literally nothing more.
Performance enhancements have been an interesting sidebar however.