Essential cookware: starting from scratch
there are several such lists, but here I will refer just to cookware needs.
your set is an excellent choice. I suggest, however, that you do not buy any other cookware for the time being, and just live with your current set for a few months. You will discover that some pieces are junk that sit at the bottom of a shelf, gathering dust, never used even once. Others, you will use almost everyday. This is where you will discover that you might need a pot or pan that is not included in your set: this is where you spend $$$ on another piece.
Offhand, I would say that you will also need some nonstick stuff: a 2 qt saucepan (for stuff that always sticks, like oatmeal or rice), and 10 inch and one 12 inch skillet.
re: jerry i h
Good advice. Don't try to get it all at once, esp. if you start with a set. I'd recommend a nonstick skillet, the best chef's knife you can afford (handle knives before buying), a cutting board for meat and one for non meats, a cast iron skillet or two, and a decent quality cookie sheet or pizza pan. I would not want to be without a microwave oven and either a good toaster or a good toaster oven. If you end up with a sub par oven in your new digs, a convection toaster oven would be a good thing to have.
I'd also second the rec for a colander or two. And I'd want sets of measuring cups and spoons. I also would want glass liquid measuring cups.
That Tramontina triply stainless steel is repeatedly cited as the best value set. The only thing to think about is about 8 piece vs 10 piece set. The 8 piece set is much cheaper, but it is a set with smaller size cookware which can be more suitable for a starter kitchen, so you have to decide which set fits you better.
There are so many things you can get, but I will focus on the essentials. Aside from cookware, I say one good kitchen knife and a decent cutting board are essential. A good kitche knife with a bad cutting board is of no use and vice versa. For a traditional German style chef's knife, I say go for a Victorinox Chef's knife or a Wusthof Chef's knife. A Japanese gyuto or Chinese Chef's knife is great too, but you only need one good knife, not three.
A paring knife is almost a must as well, but if you are really good, you can actually work with most thing with the Chef's knife.
For a cutting board, an end grain wood cutting board is a great choice. You can usually grab one around $20 at discount stores like TJ Maxx or Home Goods. A rubber cutting board is excellent too and is often overlooked. A rubber board is not a normal plastic board. Many professional kitchens use rubber cutting boards. They don't look great, but they are extremely functional. Here is a quick review from NY Time.
I wrote about my experience on a rubber cutting board 2-1/2 months ago:
In reality, any wood cutting board is fine and you can get one really cheap <$5.
Needless to say utensils are important. The most important cookware utensils are a spatula, a spoon and a pair of a tong. A spatula is more important than a spoon if you have to get just one. Of course, make sure you have the normal dining flatware. You can even use them in cooking if needed. Two spoons together can act as a small tong.
There are so many other things to get. If you are into baking, then bakeware such as baking pans, mixing bowls, a mixer, a rolling pin, a pastry board ... are all essential.
Good set of measuring spoons and cups - METAL, not plastic, with etched and not silk screened volume markings
Set of strainers
dish towels (lint-free muslin are my faves)
PE cutting board
Accusharp knife sharpener
Some stirring spoons
a slotted spoon
if you drink wine, a corkscrew - I love the Trader Joe's double action corkscrew and it was only like $5
A GOOD potato peeler
a GOOD can opener
maybe a pasta spoon if you serve pasta much
Cookie sheets, the heavier duty the better
pyrex baking dishes 11 x 17 and 9 x 13
straight sided cake tins (again I prefer pyrex if you can get them with straight sides)
A roaster if you do roasts
Pie pans if you do pies (I prefer the plain pyrex ones and they're also cheap)
Guard for knife blades, like this:
(I got mine individually at a kitchen specialty shop, you don't have to buy a whole set
)Pizza cutter if you do pizza
Baking stone if you do pizza or bread
8 qt stock pot - my son bought one of these and it has come in handy much more often than I thought it would - get a heavy duty one though
Pyrex 2 and 4 c measuring cups - don't get the plastic ones, they flake, fade, and won't go through the dishwasher
Mixing bowls - be wary, a lot of things sold as mixing bowls these days are way too shallow to make good mixing bowls
FLEXIBLE Silicon spatulas, at least one "regular" size and one small enough to slip down into jars to get the last of whatever goodness was in there, out.
pastry blender - I use mine more often for breaking up cooked ground beef or smashing potatoes than anything else. Crumbling egg yolks, bacon, etc etc etc
Again, if you do roasts, a meat fork
If you bake cookies much, or bread, a Super Parchment (or 2 or 3, to line all your cookie sheets)
thermometer that goes up to at least 210F
candy/deep frying thermometer
Ice scream scoop! Mustn't forget that, very important!
Some things that are only as important as how often you think you'll use them:
hand or stand mixer