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How did the early designers of spacecrafts have any idea what space was actually like?
How was a vessel ever constructed that actually flew through Earth's atmosphere without burning up, and then was able to travel around once it reached space without just freezing up?
How did they ever speculate no gravity and train astronauts according?
This is a very broad question, but I'll take a stab at it.
It was understood that gravity pulled the Earth into a spherical shape, with dense solids and liquids below less-dense gases, and it was expected as early as the 17th century that the atmosphere would get steadily thinner with increasing altitude, giving way to vacuum. These expectations were confirmed with e.g. balloon observations as @ACAC notes.
Vacuum chambers on the ground were straightforward to make, so predictions about heating and cooling of a vehicle in the vacuum of space could be confirmed.
Heating due to high speeds in air was also understood from the 1920s, and the extremely high temperatures encountered during reentry into the atmosphere from near-orbital speeds were studied with ballistic missiles through the 1950s.
Flying ballistic trajectories in aircraft simulated weightlessness for short periods, allowing astronauts to prepare somewhat for the experience.
One important note is that rocketry predates space travel - by a lot. The V-2 rocket christened "MW 18014", the very first suborbital flight, (meaning the rocket flew to space, but didn't go fast enough to orbit the earth, so came back down) was in 1944. This rocket was designed to launch, fly in a specific trajectory, and land in a (relatively) specific location - and it worked. It didn't "do" anything, it had no practical payload. But it worked.
By 1944 our knowledge of physics/astrophysics was fairly advanced - we understood quite well that orbits are just a matter of going really fast around a celestial body. So, armed with a rocket that can fly into space, and knowledge of how orbits are made, spaceflight was a very natural next step.
It took quite a few tests like the one linked above to understand the engineering behind the first rockets, but for the most part rockets' trajectories, the physics of space, and the nature of vacuums were already known. It was just that these things were mostly known by theoretical scientists - not by engineers. So the first spaceflights were mostly just applying the theoretical understanding we already had of how an object might go from the ground to an orbit around earth.
It should be noted that while I make it sound simple, the actual engineering that went into making real spacecraft was very, very hard. To more directly answer your question about how humans found out how to deal with the unique challenges of space travel; we iterated for a long time. It took 13 years between MW 18014 and the successful flight of Sputnik 1, which was the first manmade object to orbit the Earth. 13 years is a long time to iterate on a design - even accounting for the chaotic political effects of the end of WW2 and the diffusion of German knowledge after it.
The way we learned about how to make a craft that could take a person into orbit was by actually sending objects up to space; rocket scientists were simply launching more and more satellites into the atmosphere to study the environment. By the time Yuri Gagarin became the first man in space in 1961 - a good four years after Sputnik - scientists already had a lot of experience sending rockets into orbit and dealing with the crazy effects of space travel.
tl;dr the way people learned how to deal with space was by shooting rockets into it until they figured out how it worked. Then they shot people into it.
I think this part of your question wasn't addressed very well so far:
How did they ever speculate no gravity and train astronauts according?
Russell Borogove already said a word on the second part so I'm gonna add something to the first part.
Actually there is quite a lot of gravity near Earth. For example gravity on the ISS is still around 0.89g meaning that it accelerates towards earth at 89% of the rate we have down on earth's surface.
The laws of gravity were known as early as 1687 from Newton's law of universal gravitation and Einstein's famous general theory of relativity gave additional insight in 1915 (See @vsz comment). So we knew that as soon as we shoot a rocket to ~400km above ground it would still accelerate downward at ~8.7 m/s^2 (increasing with reduced altitude). So in order to stay at constant height (=orbit) we need to account for that somehow. This means we knew it would be zero acceleration towards earth because that is exactly how we need to design the orbit in order to stay in space and not crash into the earth.
The problem now is: how to cancel out gravity. One explanation is that on observer on earth would see the space ship moving side ways so fast that it literally falls around earth (and this observer wouldn't say there's no acceleration on the space ship !). For an observer in the space ship we could calculate that (for a circular orbit) a centrifugal force is accelerating us away from earth at the same rate gravity attracts us which means there is an effective 0g acting on us but only in this rotating reference system.
TL;DR: Gravity still pulls us towards earth but we must counteract it in order to not fall down. On a space ship that would not rotate around earth but fire its engine to keep a constant low altitude we would feel almost as heavy as on earth.
By using science to make educated guesses, by performing tests on the ground as much as possible, then by attempting actual flights and learning what works and what doesn't, and then improving and taking on more ambitious goals. The other answers have gone into more detail but I want to share the following link which I think is extremely illustrative of the incremental nature of space flight and engineering.
http://mentallandscape.com/V_Venus.htm
It covers the many Soviet missions to Venus, starting with missions that went as far as the moon, then some more missions getting into interplanetary space, then ones that actually make it near to Venus, then ones that actually hit Venus, then missions which try to make it through the atmosphere, finally cumulating in a landing and a photo from the surface. It is long but an excellent read, and shows how each mission builds on the last, and become increasingly more complicated and ambitious.
