First, your assumption is wrong for the simple reason we need nearly 8km/s to bring any payload - and all the fuel to accelerate it any further - to LEO. And that means burning enormous amounts of fuel burned during the launch phase; for every kilogram of fuel in orbit you need several kilograms of fuel burned in the atmosphere. This is known as the Tyranny of Rocket Equation, and this is an unavoidable, fixed initial cost to get anything into space - and bring it up there fast, because on top of all the acceleration you give your payload to bring it to orbital speed, every second you're losing 9.8m/s of that speed to "gravitational drag", fighting Earth's pull to keep from falling.
But once in orbit, all the fun begins with orbital mechanics. And there's a lot of savings and caveats there.
Your primary saving is the Oberth Effect. There were many questions on this site about understanding it, and I encourage you to look them up. The essence is that if your fuel has a high kinetic energy (that is - it's moving fast; the ship is flying fast) - then by burning it, you harness this energy into more acceleration.
Orbital mechanics works in such a way, that a body in elliptical orbit moves faster when it's close to the body it's orbiting - so there's more to be gained by burning in LEO than any farther.
OTOH direction corrections are easier at low speeds - near apoapsis of an orbit.
And then there are gravitational slingshots; using other bodies (planets, moons) to steal a part of their kinetic energy, entirely for free.
I can encourage you to play Kerbal Space Program if you want to explore all these intricacies in an intuitive way, not through dry equations but more like a gravitational theme park :)
But to give a "tl;dr" answer: the orbit around a single mass always has the same amount of total energy (Kinetic + Potential). This is true, but only the Kinetic energy can be usefully transformed into speed by spacecraft engines. Therefore burns performed where the Kinetic component is highest are most efficient.