What size USB stick do you recommend? What level hardware requirements do I need?
USB Stick: I think with an 8GB stick, you should be able to fit ANY distribution. Most would fit on a 4GB or smaller stick.
System requirements: it varies by which distribution you use. For example, on the low end, the "Raspbian" distribution is designed to run on Raspberry PI hardware (small credit card sized computers)...specs are basically the same as a smartphone about 4 years ago.
On the other hand Fedora (based on Red Hat) requires a 1GB processor, 1GB of RAM, and 10GB of hard drive. (and really needs about 3 times that size to really run well).
The nice thing with a live edition on a USB stick is that you can play with it to see if your system works well with that distribution: if it doesn't, then you can simply create a different live USB stick with a different distro and try again.
A couple of wiki pages that have good comparisons between distributions: