For newer old computers I would suggest, in no particular order: Vector Linux, WattOS, Puppy Linux, Trisquel Mini, PeppermintOS, SliTaz. From those, WattOS and PeppermintOS are most likely to be usable for a Linux newbie without too much problems.
Vector Linux is good, if you can configure it to your liking, which might take a while on the first time. Puppy is pretty good overall, but might be a bit confusing if you are not willing to learn how it works.
Both WattOS and PeppermintOS are based on Ubuntu, so there is a bigger chance that in case of problems the tips and advice given to Ubuntu users will apply pretty much straight to them.
For computers slower than Pentium III, Tiny Core might be the best bet, but it can be confusing to setup without reading a lot of tutorials, I would give Puppy a try there too, and see if it runs well.
Q4OS, now that is a distro that I haven't tried ever myself, and haven't even heard of it before, might have to try it out in the near future.
Sad thing is, that while modern Linux can be made pretty lightweight, Pentium II and lower computers might do better with older versions of Linux, but that ofcourse means no new updates or programs.
The slowest computer I have running at the moment is an Asus EEEPC 701 SD which has 8GB "SSD" Mobile Celeron 900MHz processor, which I think is underclocked to 630MHz, 512Mb or DDR2 ram and Intel GMA900 GPU, I use WattOS on that thing, and while it is a bit slow using Firefox thanks to the low amount of ram and really totally crap processor in all other ways it is surprisingly usable. Even Firefox works well enough, if you stay out of sites like YouTube and such.
EDIT: The most important thing I have found playing around with these lightweight Linux distros, is the amount of RAM, if your computer supports 512MB or more RAM, and you can install that much, you will have a much better time on old computers there. The OS itself runs fine on say 128MB, but the software, if you want to go as modern as possible, will hog most of it and swap like mad.