Space

Well obviously gravity has infinite range in principle, however it does fall off with the square of the distance. So it will get irrelevant at some point, with local sources being the major factor. But obviously the Earth's influence is still quite significant, since the Moon doesn't go flying off. The Moon's orbit is what it is because of the gravity of Earth, although the Sun is a huge factor as well.

The Moon's gravity is what's causing the tides on Earth, so both objects influence each other through gravity signifanctly. The spacecraft will be decelerating on the way to the Moon because the Earth is pulling it down. However once they cross a certain point, the Moon's gravity becomes dominant and they get pulled towards the Moon. They are going too fast to be able to be captured by the Moon and get into an orbit. But they are going too fast to be captured by the Moon. So they will sling past the Moon, the Moon will pull them around and they will fall back to Earth again. They would then be going too fast to be in a proper orbit, instead being flung past the other side. So they will perform a braking maneuver to go back into Earth orbit and slow down further still to land back on Earth.

What confuses people is they see astronauts at 300km up and they are weightless. So one would assume gravity falls off much faster than it does. In fact at 300km the gravity from Earth is still 90% of what it is on the surface. The reason those astronauts are weightless is because they are falling.

Imagine a skydiver falling down and shaking hands with another skydiver, they are weightless compared to each other. If it weren't for the air rushing by, you would be able to tell they were falling and they would look weightless. In fact there is this awesome video floating around where skydiver also have a car falling down and they get in it, it all looks weightless, because it is. The famous vomit comet plane does the same, it flies up, then follows a parabolic curve down (aka falling), before pulling up again. Whilst falling things are in fact weightless.

So if those astronauts are falling, how come they don't fall down to the ground, obviously something is keeping them up there. The answer is speed, they are falling down, but at the same time going horizontally at great speed. Such a great speed in fact, they miss the Earth and instead fall around it. Imagine shooting a cannon faster and faster till it goes over the horizon and if fired fast enough all around the world.

Going to space isn't about going up, it's about going fast. The reason rockets go straight up is because they want to get above much of the atmosphere as fast as possible. This reduces air resistance and makes going faster easier (or possible even). As soon as they go up a little bit, they start turning to go horizontally, often in a so called gravity turn. This is where you use the gravity of Earth to curve downwards, so the gravity helps out instead of fighting it. LEO is only something like 300km up, and the widely recognized limit for "space" is only 100km up. You could easily drive that with your car in no time at all, so why do we need a huge rocket to get to space? Because to go into orbit you need to be going 8km/s, not 8km/h, but 8km every second. To go that fast you need a lot of power and a rocket has that power.

Orbital mechanics are really weird when you first learn about them. But we have the great Kerbal Space Program game to get a feel for them. Speed and altitude are directly related, changing one influences the other, often in ways we feel are not intuitive. This is why people who know faceplam at a movie like Gravity where an escape pod points up and just flies off into deep space. That's not how any of this works, most likely there wouldn't be enough energy (delta V as the nerds like to say) to get out of Earth orbit. At best it might end up in a helio centric orbit near Earth, but it's unlikely. It isn't like you can just point to outer space and go there.