What type of skills are needed to create your own client-side, server-side types of coding languages?

Programming language design is its own subject, and you could probably find some literature or essays targeting it.

A fluency in a compiled language is a skill definitely needed, C or C++ being the obvious choices.
You could write a proof of concept if you so desired in an interpreted language such as php, but the real time use of that is below zero, still at least you could see how your language could be translated.

A language parser 'simply' translates your code into machine instructions, easier said than done.

If written in C / C++, Flex and Bison should sort your lexical scanning and language parsing needs; if you were feeling hardcore, you could write your own.

The language is then integrated into the environment. For a server side language, the majority of web servers provide an API for writing modules, for example Apache does. If you were to write a client side language, you would have to contact the vendors of that browser and propose they integrate it into their browser, which seriously is never going to be an easy feat, ECMA / Javascript is widely accepted and there is no room (or need) for anything else. Unless you mean client side in relation to scripts that run on the client machine, then all you have to do is provide a binary for the user to download, which is your parser.

Here's some useful links:

http://en.wikipedia.org/wiki/Tokenizing
http://modules.apache.org/doc/API.html
http://cs.uic.edu/~spopuri/cparser.html
PHP Source : http://www.php.net/get/php-5.2.4.tar.bz2/from/a/mirror

If on the subject of System Programming, you can customize C++ to quite an extent to make it look like a completely different language, keeping all the characteristics and usage the same. Building your own input output streams isn't too difficult and many people use it when they build private programs, although not recommended for a large open source project.