I have 24 GroVia wipes and could do with some more really for safety. They are a great size and texture so do the job really well. I store them in just a normal snap-down lock Tupperware box (iykwim).
For face and general wipe downs I just have some small thin flannels, but recently used a friends one and I'm totally sold as did a much better job on my now very mucky baby led weened 10 month old! It was a Little Lamb wipe but was really big and fairly thick, I think like a bamboo flannel?
SmartiPants do a SmartPail wet bag which is a zip lock (which I find much better than drawstring) and is a really good size at very reasonable price. I haven't had any leaks and even though one of them has delaminated a bit this doesn't seem to have effected the bag overall. They also do smaller ones for out and about. They are just plain colours but if you're looking for cheep and cheerful it's a good choice.
I also have a Bubba & Boo wet bag for out and about and then my 'treat' one was a Blueberry wet bag.
They all work just as well as each other but the Blueberry ones come in great prints
I must say that even though it was a bit more expensive the Blueberry wet bag is definitely a bit thicker and more robust feeling.
Have a look on the Thenappygurus.com site cos they have loads at good prices.