Here are some things I find extremely useful:
12" cast iron skillet
8" cast iron skillet
big stock pot
bamboo stirring stick
metal spatula/burger flipper
colander (bonus if it fits into your stockpot)
Chinese cleaver (a thinner vegetable one rather than the bone-crushing ones)
some method of sharpening the knives
wood cutting board
second cutting board if you do meat/chicken/fish
big Chinese bowls- good for prepped veggies, eating cereal, soup etc.
a couple of even bigger bowls for mixing things
Mason jars for food storage
A lot of this stuff can be cheap, cheap, cheap. You probably already have a stock pot and a saucepan in your set, so no worries there. The cast iron stuff takes a little extra care, but it is inexpensive, works very well, as is versatile. You can bake in it, too. Basically, you need a big one and a little one. I measured the top diameter of mine, not the base. Anyway, 8" versus 12" makes it sound like they're close in size, but they're really not. Lodge is the only remaining US maker of cast iron, and their stuff is good and comes pre-seasoned. Make sure to keep hot pads near the stove. Cast iron teaches you to use them, because the handles get hot! It is nice to have lids for your cast iron skillets. If some of your existing lids fit the new skillets, then you're good to go. The cast iron lids are heavy and expensive.
I like Chinese cleavers. Most folks like Chef's knives. People say good things about Victorinox/Forschner knives. Like the skillets, just get a big one (for most things) and a small one (for detail work). Make sure you have some method for sharpening. Self-adhesive sandpaper of various grits stuck to mousepads is supposedly an effective way to get knives "scary sharp." I used a Crock Stick for years, and am now moving to Japanese waterstones, but there is a learning curve for that, and they're not cheap.
A big cutting board makes things easier. Stay away from cutting boards made of glass or hard plastic or any kind of stone. They'll dull the knives.
I like Mason jars for food storage. They don't work with everything, of course (think- leftover pizza!), but for the things that do work for, they work very well because the glass doesn't retain odors or warp out of shape in the dishwasher.
Hope this is helpful!