Speed isn't needed to escape Earth's atmosphere. Helium balloons have escaped the overwhelming majority of Earth's atmosphere.
I watched some YouTube videos, but I'm confused about why speed is given versus distance. This brings me to orbit. In miles, what is the closest safe orbit (safe from quick orbital decay)? You speak of the ladder. Must the direction be away from the ladder vertically or perpendicular to the ladder?
Perpendicular, sort of. It doesn't have to be precisely 90 degrees, but the closer to 90 degrees the better, and how close to 90 degrees you need to be depends on how high up you are -- at 100 miles up you need to be very close to 90 degrees but at 1000 miles up you can get away with a lot of slop. John Glenn was 100 miles up, which is about as low as it's safe to orbit.
The simplest way to understand why escape velocity depends on distance is to imagine the sun and the other planets gone. Go out ridiculously far away, say a billion miles, and drop a rock. Because the Earth's gravity is next to nothing at that distance, the rock will fall to Earth incredibly slowly, taking millions of years. But gravity gets stronger as you get close to Earth, and the rock will speed up a lot in the last few days just before it hits. It will hit the atmosphere at 25,000 mph. When the rock is still, say, 22,000 miles away, it won't be going as fast as that, because it's still got 22,000 more miles of falling and speeding up to do. It will only have sped up to about 10,000 mph. The point is, "escape velocity" is just the name for running that whole scenario in reverse. The laws of physics are almost completely symmetrical, which means if you throw a rock straight up from the top of the atmosphere at 25,000 miles per hour, an hour and a half or so later when it's 22,000 miles up, it will have slowed down by 15,000 mph -- it will have slowed down exactly as much as the dropped rock would speed up during that last 22,000 miles of fall. Up or down, the 10,000 mph it's going when it's 22,000 miles up is the Earth's escape velocity for 22,000 miles above the ground.
Once you understand escape velocity, orbital velocity is simple. The speed of a circular orbit at any given height is just the escape velocity at that height divided by the square root of 2, about 1.4. So the speed of a low circular orbit is 25,000 / 1.4, about 18,000 mph. At any speed below that you won't be able to maintain orbit at that height -- you'll fall into a lower orbit or reenter the atmosphere. At any speed intermediate between circular orbital velocity and escape velocity, you'll get an elliptical orbit that repeatedly carries you up further into space than your starting point and then brings you back down.
Which brings us back to your original question about a possible future technological breakthrough that counteracts the need for incredible speeds. Yes, we know of one. It's called a "space elevator". Imagine we put a big space station in orbit 22,000 miles up. Escape velocity is only 10,000 mph at that height. So imagine we dangle a 22,000 mile long cable from the space station all the way down to the surface of the Earth. Then you don't have to ever go 25,000 mph to escape from the Earth. You just grab onto the cable and climb up as slowly as you like. You can build a cable car that grips the cable with drive wheels, and get into space using an electric motor to climb at 22 mph for 1000 hours. Then you can launch your rocket to Mars at only 10,000 mph instead of at 25,000 mph. It's even better if all you want is to get into orbit -- then you only have to launch at 10,000 / 1.4 = 7,000 mph. And here's the wonderful bit: the cable is hanging from a space station that's already in orbit, which means that by the time you get to the top of the cable, you're already going 7,000 mph. You've just spent the last 1000 hours gently speeding up to 7,000 mph at an acceleration of 7 miles per hour per hour -- a g-force of about 1/10,000th. When you get to the top you can get into orbit just by letting go of the cable.
Now here's the downside, the part that means we need a technological breakthrough. When you're 22,000 miles up, and you're going 7,000 mph in a circle around the Earth, one orbit will take 24 hours. That means the space station hovers over one spot over the equator as the Earth turns below it. That means the bottom of the cable can be anchored to the ground instead of dragging along the surface at hundreds or thousands of miles per hour, which would make it tricky to grab on. I.e., you can't do this space age Indian rope trick from only a hundred miles up, because the space station will orbit too fast. The cable really has to be 22,000 miles long. Nobody knows how to make a cable 22,000 miles long that won't break from its own weight.
