Sorry I would have commented sooner but I had a crazy busy day.
First of all do not try to understand relativity without understanding Newton’s laws of motion. It would be like trying to study Kaballah without knowing Tanach first. (though with more math) Every physics department in the world starts by teaching Newtonian physics, I know mine did.
One important point about both SR and GR is that they do not get rid of Newtonian mechanics, but extend it. Newtonian mechanics are a special case of relativity. In the case where the object is moving slowly or not near a VERY large gravity field Newtonain mechanics and relativity give EXACTLY the same results. In terms of the solar system the only place where they give different results is in terms of the precession of the perihelion of the planet mercury, and that effect is measured in 10’s of arc-seconds per century.
It terms of time dilation the amount time slows down on a moving body is with a factor of the sqrt(1-v^2/c^2). So the closer you can get to the speed of light the easier that is to measure. In terms of subatomic particles which can get VERY close to c it is quite easy to measure. In terms of more macroscopic particles it is harder. But the best atomic clocks can be accurate to 1 part in 10^14, so for example an atomic clock on a GPS sat in orbit will not run the same as a clock on the ground. The effect is small but it is there.
GR came about when Einstein decided to ask a very simple question, he noticed that gravitational mass and inertial mass are always the same and wondered why that should be so. He then spent 10 years working out the math to turn it into something that you could actually use, and missed the chance to predict the big bang 15 years before Edwin Hubble found it.
If people have specific questions please ask them and I will try to respond (or ask someone who would know)