With the original XBox system, you could NOT change your XBox Live gametag. It was permanent. With the new XBox 360 Live system, you CAN change your gametag. It is not free, however - it costs 800 Microsoft Points.
So step 1 is to get 800 Microsoft Points into your account :) Log into your existing Live account and use the points area to add in some money with a credit card.
What's vastly interesting is that Microsoft had an "exchange rate" on their initial launch - and they haven't changed it since. So since actual real life exchange rates have been going up and down over time, people in some countries are paying much more than they should for their points!
For example, at the beginning you could get 80 points for $1 - or the equivalent of $1 in other countries. Everything was even. You bought points in blocks of 100, so in the US a block of 100 would be $1.25.
In modern times, when a person in the UK buys 100 points, they are paying the equivalent of $1.75. Someone in Canada is paying the equivalent of $1.62. Microsoft should get their online economy back in line!
So, anyway, if you're getting 800 points, that is $10. So you're paying $10 for Microsoft to do the hassle of changing around your gamer tag ID. I don't think this is too bad - if they had it free, people would be swapping their gamer tags every 10 minutes and it would bog down all of the servers and cause huge problems. At least this way someone has to be serious about fixing a typo or having another reason to switch the spelling of their gamer tag.
So once you have the money in your account, you go look at your profile, and select the edit profile option. Select the gamertag associated with the profile. There should be information right there to tell it what you want to change it to, and you confirm that you are going to pay the 800 points to do it.
This will update it with all of your save games, with your achievements, and even with all of your friends lists. Now go contact all of your friends and let them know so they don't think an invader has gotten into their friends listing!