Every site and video I have been on doesn't address it directly.specifically I want to know what is are the experiments we derive these wavelengths from and how do we calculate the wavelengths inspite of this evidence?
Great Question! We use a device called a spectrometer to separate waves of the electromagnetic spectrum and use the separation to approximate the wavelength of passing radiation. This device is commonly used to identify materials/elements as only a certain wavelengths of light will be admitted through different materials. Here are some websites to further your understanding...
Spectrometers are using various methods of splitting incident light by wavelengths in a predictable manner (where the direction of a light of specific wavelength is directly computable from its wavelength and vice-versa - a diffraction grating is one example). This is where a simple prism is not so good by itself until it is calibrated (it's a chicken and egg problem: how do you measure index of refraction at different wavelengths if you don't know the wavelength of the testing light - but after comparison with a known reference, you can use it just fine). The grating doesn't need calibration. For instance, if you know how far apart the grooves on a CD are (if you still remember what CDs were), you can measure the wavelength simply by looking at which angle a certain colour is reflected.
You could also use reference monochromatic light sources with computable wavelength. Gas emission spectra are examples of this (or a free electron laser, for the purposes of discussion).
I'd like to point out a somewhat loose and inaccurate phrasing of the question... colours don't have wavelengths. Colours are names given to visual perception of light, and there are infinitely many ways to combine different wavelengths into the same perceived colour. Most colours are not representable by monochromatic light. And those that are, can also be done differently. A conventional RGB screen makes all its colours by combining three colour components, each of them having a spectrum of its own (depending on the type of display: for LCD, it depends on the transmission spectrum of the filters and the spectrum of the backlight). Just to make things clear.