If you're going to try to get music for free, don't use Limewire or any peer to peer file sharing service. More than likely the music is pirated - you will be downloading an illegal copy, opening yourself up to a possible lawsuit.
You might be able to find some free downloads from Amazon.com and maybe even Download.com (a CNET site - click on the "downloads" tab and then "music" - some of it is streaming, some of it is available to download)
I buy almost all of my music. I have a large cd collection that I have put on my computer. I supplement this with tracks purchased from iTunes - I have an iPod. Every week, iTunes offers a couple of songs for free. Usually, they're from lesser known artists.
Since I don't know what kind of mp3 player you have, it's hard to help you. If you have a Plays For Sure device and it supports subscription services (such as Napster), you can pay a monthly fee and download all the songs you want. You won't own them, you won't be able to burn them to an audio cd, and if you stop paying you will no longer be able to play them. But it is relatively cheap if you don't have an extensive collection.
There are a couple of other subscription alternatives that will work with any player that plays mp3s - eMusic and Audio Lunchbox. You pay a monthly fee for a set number of downloads. Depending on your subscription level, songs cost as little as 33 cents each or even less (this is an eMusic claim). Audio Lunchbox works similarly. With both eMusic and Audio Lunchbox you get mp3s, you can burn them to cd, and you own these songs. They do come with a free trial.However, you won't be able to download all that you want and they tend to skew toward independent artists.
You don't need to be a computer expert to use an mp3 player, just a little motivated and be willing to spend a little bit of time on this. Whether you spend a little bit of time or a lot depends on how much music you want. It would help if we knew what kind of player you have and whether or not you already have some music - if it's on cd or your computer.