I want to make a script that changes screen brightness and, among others, need this command:
echo "$number" | sudo tee /sys/class/backlight/intel_backlight/brightness
The script asks me for my root password which i think is unnecessary for it only changes the brightness. I tried adding sudo -S and echo-ing the password but not only did i confuse myself with what input goes where, but the script writes out the [sudo] password for user: prompt which is anoying. How do i make the script runable by everyone (both from inside of the script and outside, i do this as an exercise to learn more)?
You might configure your system so that sudo does not ask for any password. I don't recommend doing this (put ALL=NOPASSWD: in your /etc/sudoers file at appropriate place), since it is a security hole.
But what you really want would be to make a setuid executable (BTW /usr/bin/sudo is itself a setuid executable). It is tricky to understand, and you can make huge mistakes (opening large security holes). Read also carefully execve(2) & Advanced Linux Programming. Spend several hours to understand the setuid thing (if you misunderstand it, you'll have security issues). See also credentials(7) & capabilities(7).
For security reasons, shell scripts cannot be made setuid. So you can code a tiny wrapper in C which would run the script thru execve after appropriate calls (e.g. to setresuid(2) and friends), compile that C program as a setuid executable (so chown root and chmod u+s your executable). In your particular case you don't even need to code a C program starting a shell command (you just should fopen the /sys/class/backlight/intel_backlight/brightness pseudo-file then fprintf into it, and fclose it).
Actually, I don't believe that doing all that is necessary, because you should be able to configure your system to let your screen brightness be set by non root. I have no idea how to do that precisely (but that is a different question).