Sign Up for a Second Life Avatar
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You choose a first name, select from one of a predefined (and changing) list of second names from a drop down box, and then check if that is unique. Name capitalisation sticks with you, so unless you really mean to have a lower case start to your first name, make sure you capitalise the name the way you want it to appear.
During signup, I suggest you skip the step to "Join a Community", and then you will go to the generic orientation welcome area when you first log on to Second Life.
Owning land to build things on such that they do not disappear and buying nicer things for your avatar to wear (you will find this essential after you get used to Second Life) costs a little through the in-world currency - Linden Dollars.
You don't need to give billing information for a free account. But if you do it gets you a few starter $L - you can just connect your PayPal account or give credit card details at the initial sign up stage and you will get a few Lindens to give you a stating position... yet it still not cost you anything unless you later upgrade you account or buy $L... which would then be charged through your nominated payment mechanism.
One way to get a good value paid account is to take out a subscription on an annual basis. This costs less per month, and gives you a weekly stipend in $L which returns a large element of what you pay for the subscription.
Second Life requires a reasonably powerful computer - Windows, Apple Mac or Linux versions are all available. It is memory and graphics card usage intensive. See the System Requirements page.