What are the most important PHP WordPress functions you must use to make a good website or whats Templates classes and functions you always useORmake?

For making themes from scratch, the best choices out there in my opinion are:

1. Underscore (starter theme)
2. http://roots.io/
3. Gamby Framework

Those three good to haves for any serious WordPress developer/themer.

As for the site itself, I strongly recommend to put some effort in defining security and backup policies from the very begining, before starting to optimize the website for speed, some good practices are:

1. Do automatic backup, the more frequent the better
2. Do not use outdated version of WordPress
3. Do not use software (PHP5.2 or5.3, etc..)

Those are really the minimum.