I personally use Ubuntu and reccomend it for a new linux user. It is relatively small compared to multi cd distros like red hat and mandriva,and includes lots of graphical tools for system configuration, and adding new software is as easy as opening synaptic or using apt-get from the command line. It even tells you when you have software updates!!
For older computers i would suggest Damn small Linux.You can run it from a flash device and it all loads into ram, will run on anything provided the video card is supported, also with good configuration tools,and again adding software is easy.
Debian is really my os of choice because the base install is so small and you can add only what you need, but not as difficult to install as Gentoo,which i really wanna get into.I use ubuntu only because debian takes a bit more configuration and lacks some of the graphical config tools.