Which characters you must and which you mustn't escape indeed depends on the regex flavor you're working with.
For PCRE, and most other so-called Perl-compatible flavors, escape these outside character classes:
.^$*+?()[{\
and these inside character classes:
^-]\
For POSIX extended regexes (ERE), escape these outside character classes (same as PCRE):
.^$*+?()[{\
Escaping any other characters is an error with POSIX ERE.
Inside character classes, the backslash is a literal character in POSIX regular expressions. You cannot use it to escape anything. You have to use "clever placement" if you want to include character class metacharacters as literals. Put the ^ anywhere except at the start, the ] at the start, and the - at the start or the end of the character class to match these literally, e.g.:
[]^-]
In POSIX basic regular expressions (BRE), these are metacharacters that you need to escape to suppress their meaning:
.^$*
Escaping parentheses and curly brackets in BREs gives them the special meaning their unescaped versions have in EREs. Some implementations (e.g. GNU) also give special meaning to other characters when escaped, such as \? and +. Escaping a character other than .^$*(){} is normally an error with BREs.
Inside character classes, BREs follow the same rule as EREs.
If all this makes your head spin, grab a copy of RegexBuddy. On the Create tab, click Insert Token, and then Literal. RegexBuddy will add escapes as needed.