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There is a few different concepts here in conflict, which is why the question is hard to answer.
What do we mean when we say space? Usually we mean above the Kármán line, or above 100km. At that point you are above almost all of the atmosphere, so we consider that space. The atmosphere does actually extend quite a bit above that, but at that point it's so thin we consider it to be space.
However as we know, the Earth has a bunch of gravity from its mass. So when you get up to 100km you just fall back down. Space isn't free of gravity, the gravity of the Earth extends basically forever. It's influence does get less (thanks to Newton we know by how much), but considering the Earth is thousands of km wide when we get up to 100km we are pretty much still on the ground as far as gravity is concerned.
So why do we see astronauts fly around? They are weightless, so there is no gravity right? This is something popular media gets wrong a whole bunch, it's not like there is some magically line called space and beyond there you are weightless. Those astronauts are actually in orbit, that's why they don't experience gravity from the Earth. To understand orbits, imagine we fire a big ass cannon. The ball flies through the air in an arc and lands on the ground. How far away it lands, depends on how fast we shot the ball. The faster it went out of the cannon, the further it flies. Now imagine we shoot the ball over the horizon, so it lands so far away we can't even see it anymore. It still lands right? Yes, but only up to a point. It turns out if you shoot the ball fast enough, the arc just continues falling beyond the horizon until it loops around the Earth. As it is falling, it doesn't experience gravity except for the arc it follows.
Usually when we put stuff into space, we mean put it in orbit and especially something called Low Earth Orbit . That means it needs to have a speed just like the cannon ball, to keep falling indefinitely. The speed we need is dependent on how large the arc we want to have, or in other words how high the orbit is above the Earth. For context, if we want to fly in orbit in space so at an altitude of 100km, we would need to go almost 28254 km/h. Imagine driving that fast on the highway, it's crazy fast.
That's why we use rockets, it's not as much about going up, it's more about going really fast. So a rocket takes off and goes vertical for the first bit, this is to get to a thinner part of the atmosphere to reduce drag. Then it does something called the pitch over maneuver, usually in the form of a gravity turn. This is to go mostly horizontal and get that speed up. At the speeds rockets are going, they get to the 100km altitude in no time. So they pitch over as to not overshoot and use all their energy to go as fast as they can horizontally and thus into orbit. Then you get into the realm of orbital mechanics, which popular media also gets wrong a whole bunch. You can't just point your spacecraft into space, give it a boost and be flying off into the void forever. If you want to learn more I would recommend playing Kerbal Space Program, to get a feel for how orbits work.
But say we are totally done with Earth and just want to leave it all behind, go into Deep Space. How would we do that? For that we need even more speed, something called escape velocity. If we get to that speed (40270 km/h), we can leave the Earth and go wherever we want, right? No not just yet, we might have left Earth behind, but we are still in orbit around the Sun. So we are still following orbital mechanics, only the Sun is the primary body we have to account for instead of the Earth. We can use orbital mechanics to fly around the solar system.
If we want to leave the solar system, we would need to go even faster. But the issue is there is nothing out there. To get anywhere interesting, we would need to travel close to the speed of light for years. Even our fastest spacecraft are standing still compared to the speed of light, so leaving the solar system isn't very useful right now. But we do have the Voyager space probes which kinda sorta left the solar system and we got some interesting data from them, which is cool!
This seems like a very well-informed post, was definitely full of useful, accessible information, well linked, and well written. I actually learned something (the sideways thing)! Thank you!
You are very welcome! <3