Bash comparison and expression operators

• test is a shell built-in which (obviously) tests for some given condition. Some older shells need a special command. So you'll find also a program at /usr/bin/test.
• [ is also a program to test for some condition. This software needs also a closing bracket and you can find it at /usr/bin/[.
• [[…]] is an alternative to test and [. It was developed for the Korn shell (ksh). But you also find it in Bash versions greater than 2 and in the Z Shell. The double brackets have some nifty features:
  • The shell does no word splitting or file name expansion.
  • You need no quoting.
  • Instead of -a (AND) or -o (OR) you can use && or ||.
  • The = can do a lot more.
• ((…)) is equivalent to let. So basically ((expression)) is the same as let "expression". However with let you can use more than one expression, but double braces only allow one expression.
• $((…)) (and also $[…]) does some calculation. You shell tries to calculate the expression inside the braces and replaces the expression with the result. So echo $((1+1)) leads to echo 2. So your shells prints the number 2.

Every time when you want to compare two numbers you should use the option with a dash and two letters (-ge, -lt etc.). If you want to compare strings you should use = or !=.

Your question regarding escaping is quite hard to answer. Because it depends on the software you use besides from the shell. So i.e. grep and grep -E need different escaping. This is also the case with sed, awk and others. So the best option is to have a look at the manpage. After some time you get accustomed and know when to use escaping.