I'm not sure why Ubuntu would seem unstable since supposed to be among the more stable desktop distros, at least below the hood. I haven't used it since version 11 or 12 though so I may be wrong.
Debian Stable is a popular choice for servers and is pretty rock-solid, but a lot of the software in its repository is pretty old. Fedora is also very stable and another good choice for servers since it's basically just upstream RHEL which is a commercial product, and has much more modern software due to being a rolling release.
Actually most of the popular distros are pretty stable. If you just want a nice user experience steer clear of the more hobbyist-oriented ones, though if you really want to learn how Linux works you can try Arch or my favorite, Slackware, which are both stable and easy to configure but will require you to learn to use the command line and can easily be screwed up if you aren't careful.
System resource use is usually more an issue with your desktop environment and/or window manager and stuff than under the hood. If you're don't want to hack together a UI with a WM and some other custom stuff I suggest switching to xfce or LXDE instead of whatever flashy crap your distro might come with out of the box.