A small backpack is probably the easiest bag to pack for the zoo. You can stick everything in it and it's easy for you to wear on your back, leaving your hands free for taking photos or feeding animals. If you only have a few little things to carry, a waterproof phone pouch is large enough to carry your phone, ID, and some cash, plus a locker key if you're at a zoo that uses lockers. Personally, I don't like packing in fanny packs because I just don't feel like they're comfortable, but the fanny pack actually became trendy again last year, so you could fit your small items in that too.