If you’re only keeping a few cigars at a time, a ziploc or Tupperware dish that’s “humidity controlled” will be fine. If you get serious and need a good basic one, PM me and I’ll send you mine (I don’t smoke any more).
As for cigars to try, I recommend Connecticut cigars, as they’re lighter and generally smoother and easier for newbies than some of the more robust, flavorful cigars. If you can find them, get the Rocky Patel Vintage 1999–that’s a GREAT beginner cigar, and it would be the first one I’d reach for these days if I smoked.
Here’s another idea for you: check out pipe smoking. If you smoke regularly, it’s much cheaper than buying nice cigars, and your local shop probably carries some great blends that you can try. Also, you can buy a pipe with a filter, which is a bit “cleaner”/healthier.
Finally, bourbon and scotch pair the best with tobacco (IMHO), and can really enhance the smoking experience.
Hope this helps!