I wrote this intro for a friend and since then more people have found it useful:

https://xn--2-umb.com/22/approximation

If you want a proper textbook, I recommend L.N. Trefethen (2019). “Approximation Theory and Approximation Practice.” Which is available online here:

https://epubs.siam.org/doi/book/10.1137/1.9781611975949