I know that time signatures change how a song sound, but I don't know the rule, or how it does that. If a song is in 4/4 and it got changed into 6/8, I don't know how it would sound, or how much it would change, I only know it would sound different. If anybody knows what the rule is, and how I can play a song in different time signatures, without a computer, please say it.
Time signatures are used in music notation as a way to communicate metres https://en.wikipedia.org/wiki/Metre_(music). Like you said, different metres i.e. repeating basic pulse patterns sound different. Changing musical metre is deliberate artistic modication and there is no single correct way to do it. It's like a distorting Photoshop filter - after you've distorted an image, there's no way to know what it looked like before you distorted it. As a matter of fact, you can't even know for sure if it's an original picture or derived from something through a distorting filter. (though if the person in the picture has five noses, you can suspect something) The most severe distortion removes details altogether. For example, you could change something from 4/4 to 6/8 by removing everything that's on the fourth beat.
You can also change the time signature by overlaying it on a different beat, creating a polyrhythm ("polymetre"). Or by making it a tuplet. A straight-forward way to transform a 6/8 time signature to 4/4 is to play it as sextuplets over whole-notes. Different time signatures, but if the bars remain equally long in absolute time units, the listener won't notice any difference.
Here's a 4/4 time example measure
There's any number of ways to stretch and manipulate that musical idea to fit measures with 6/8 time. Which one is your favorite?
how I can play a song in different time signatures
This is a textbook exercise for organists learning how to improvise variations on a hymn tune. Note, it's an exercise. The textbooks don't dictate answers. You learn how to do this by just doing it and listening to your attempts. "Amazing Grace" is a waltz. Try to play it as a march. After a few months of such exercises, you get it.
A time signature can convey a few things -
- it tells you how many beats are in each bar
- because of this, it tells you where the starts of the bars are, which tells you where the downbeats are (which are usually played strongly)
- it gives you a clue as to what the pattern of accenting of the beats should be within each bar.
Normally, a composer chooses to write the piece in a time signature that makes the piece look easy to read, and gives the right clues as to where the strong beats are. In other words, they choose a time signature that fits the way they want the music to sound.
I know that time signatures change how a song sound
That might or might not be true. If you changed a 4/4 song to 2/4, a performer might take it as a clue to play strong downbeats every 2 beats, rather than every 4 beats, which might change the way the song sounds - although there are different ways of emphasising strong beats, which might vary per performer.
If a song is in 4/4 and you changed the time signature into 6/8 - Only a few of the original bar boundaries would be in the same place, and the performer might just end up finding that the note durations didn't fit the new time signature very well, and get confused.
Of course you might change the note durations to fit the new time signature - but there might be more than one way to do that; you'd be making a reinterpretation that wouldn't simply be the result of changing the time signature.
Playing a song "in different time signatures" isn't really a straightforward thing to achieve - you have to decide what you mean by that.
As a bad analogy - think of the song as like legs, and the time signature as like trousers. usually, the trousers are chosen to fit the legs! If you choose trousers that don't fit the legs - how are you going to make that work? By letting the trousers tear? By making the person slim down? By cutting off parts of their legs? All of those would have different results.
You might want to re-assess your idea that "time signatures change how a song sound". That's not really quite how it works; Time signatures are usually chosen to fit how the composer wants a song to sound, but they can't on their own dictate a rhythmic feel.