Welcome to ARFCOM and welcome to the shooting sports. You've found one of the best places going if you're after solid, factual information. The down side of that is you can lose entire days just reading the Q&A threads.
As to your question, there isn't much to discuss. There's only one popular rimfire cartridge, the trusty old twenty-two.
Pro's:
Twenty-twos are great for practice. You can shoot all day for pennies and you don't make half the noise so finding a place to shoot where it won't upset the neighbors is easier.
Con's:
Twenty-twos are no good for defensive use and aren't any good as hunting arms for anything bigger than a rabbit.
Everything else, including shotguns, is center fire.
Pro's:
The sky's the limit on power, range, and accuracy.
Con's:
The sky's the limit on noise, recoil and $$$.