How do we prove something?

This is the Pythagorean theorem which only be used for right angle triangles this theorem follows or can be proven from the cosine law .

a^2 = b^2 + c^2 - 2bc cosA

if the angle that subtends A = pi/2 or and integer then cosA = 0

So the equation reduces to

a^2 = b^2 + c^2 -2bc*0

a^2= b^2 + c^2

provided the angle that corresponds to A is 90degrees or pi/2.

a^2 + b^2 = c^2 Can be proven when the angle that corresponds to C is pi/2.

But in reality the really equations is a^2 = b^2 + c^2 - 2bc cosA

a^2 + b^2 = c^2 is the formula to find the sides of a right triangle. To prove that it is true just find a triangle with all sides measured and put the numbers in. For example. your triangle would have a base of 3 in, a side of 4 inches with a hypotenuse of 5 in. That would give you your above equation 3^2 + 4^2 = 5^2 → 9 + 16 = 25. So yes that is a good way of proving it.

No, you have proved nothing. The whole point of pythagoras' theorem is that in a right angled triangle the square on the hypotenuse is ALWAYS equal to the squares on the other two sides.
There are many proofs to this theorem.

The whole point of a proof is to show that something works for the general case. When you start plugging in numbers, you are only showing that it works in that case.

By saying that 3^2 + 4^2 = 5^2 you are only showing that there exists a triangle such that a^2 + b^2 = c^2 is true.
The point is that FOR ALL right angled triangles, a^2 + b^2 = c^2 is true.

Well, if you were to substitute every possible set of numbers that would prove it but it would be a lot of work. You need to find something more general. Look up proof of Pythagorean theorem on the web to see how it is done.