Good job that you can diferentiate between the two terms
changing your IP, and
hiding your IP - You cannot change your IP. However, there are several methods open to you for hiding it.
A shell account can host a Bouncer (BNC) which can provide you with a vhost (virtual host) which is what those hostnames such as "i.h4x0r3d.micros0ft.because.3y3.4m.l337.0rg" are - vhosts. Shell accounts are not IRC servers themselves. IRCds are, in a sense, the IRC server itself - IRCds can be run on shell accounts too, so that the server remains online 24/7. You could probably obtain more information off of a shell company.
A second way would be to use a network that supports a usermode which hides your IP with a set vhost. This usermode is usually +x. Some of the bigger networks that support such a mode are Undernet and Quakenet. You must register an account with them and then login and set +x on yourself, where you then get given a custom vhost - for example, with Undernet, you get "username.users.undernet.org" where 'username' is the username you registered with. Ask in your network's #Help channel to see if they support this.
A third way would be to use a proxy. This is a more much complicated way and I have no personal experience on the issue. In general it is not the best route to go. Almost all networks do scan you for 'open' proxies when you connect and due to insecurities will ban you automatically for connecting with them. You can find proxy settings in mIRC options:
ALT+O > Connect > Firewall. Do not play with this if you don't know what you're doing.
This page may be of some help.
Hope this has been informative

Regards,