If you know exactly what you want or are adding to a pre-existing set it's good advice to buy separates -- but you don't and you aren't so it probably isn't. If you're buying an entire set, you can save a lot buying most of it as a set -- and adding to it as needed. One of the nice things about a 7+ piece set is that they come with everything you need to get you started, including lids. Don't forget lids.
Buying separates costs a fortune -- and having the best single choice of everything for your "core set" of three or four pans and two or three skillets is dumb. Everyone needs a few of the same sizes in their home set, you might as well buy them in a single box and reap the savings. At a certain level of quality they all cook pretty much the same.
You don't need the very best cookware to be a good cook, but it helps a lot if it exceeds a minimal threshold of quality in terms of responsiveness, holding the heat, spreading it evenly and so on. You already seem to have absorbed the lesson that too cheap is not good. So, enough said on the subject.
Stainless pots and sauce pans are all well and good. But stainless skillets are another matter. You absolutely want a stainless lining (or something else non-reactive like enamel-over-cast-iron) for your "core set," but stainless multi-ply is not the best choice for anything which wants a slippery pan (like eggs and pancackes), for searing and developing fond (steaks, etc.), or for holding the heat (frying chicken). Eventually you'll probably add pieces. How soon "eventually" looks like it will come can help determine whether you go for a really huge set now, or keep it down with the idea of quickly adding to it later.
In addition to the stainless lined skillets of our core set we have three carbon steel "Mineral" skillets and two cast iron skillets. A lot of pans, but they all get used.
The first thing you should add -- probably right from jump street if you don't already have one -- is a decent quality, stainless "mutli-pot" set (aka spaghetti/stock pot with two steamer inserts). You don't need multi-ply for stock pots -- a decent sandwich bottom is good enough. Why? Because with a lot of liquid in the pot, the liquid itself will evenly spread the heat so "hot spots" are irrelevant.
Look for something which feels comfortable in the hand and isn't too heavy to lift and handle. Women especially tend to appreciate "helper handles" on the larger pieces. The uber quality sets often run quite heavy. We replaced our core set with Mauviel M'Heritage 250 (stainless lined copper) which is extremely heavy. Not only are the large pans on the ragged edge of my ability to handle one handed, but they take a long time to preheat -- even though they're copper. My wife works around the weight, but -- even though I'm strong enough -- it took me a long while to get used to it.
Speaking of copper... Don't waste your money on copper as one of the metal layers in a multi-ply sandwich. It really doesn't make a difference in performance. If you want copper pans you would have said so, but for the benefit of others -- don't buy it unless you're buying it for its beauty. They're a lot of extra bucks for no extra bang worth mentioning.
Multi-ply, whether tri-ply or more layers, is very desirable. Stick with it for your reasonably sized pieces.
SLT is currently having a huge sale on its own line of multi-ply. They're good pieces -- but by no means the best or the most heavy duty. Calphalpon and Tramontina were already mentioned as two other budget lines.
Stepping up in class, I like Vollrath's tri-ply Tribute line quite a bit; but they were too industrial and ugly for my wife. We have a few pieces of the old All-Clad line (which I believe All-Clad may be in the process of discontinuing in favor of it's new "Tri-Ply") and they're very good -- they also happen to be tri-ply; stainless, over aluminum, over stainless. Vollrath and "old" All-Clad are my top choices for multi-ply stainless and, as it happens, both are made in the USA.
There's a lot of quality out there from Europe, including Paderno, ScanPan, Sitram, Matfer-Bourgeat, Mauviel, Demeyer, and more. Look around, find something which appeals, and if you have questions run them by us here. If it's a scam (like "waterless cookware," vastly overpriced, or just doesn't have a good reputation (like La Creuset stainless) someone here will let you know.
Bed Bath and Beyond is a good place to buy from, because of their customer service and generous return policies. SLT and WS (Williams Sonoma) offer some significant sales on top line cookware at this time of the year -- if you want to go high-end. Give January a couple of weeks and take a look at places like Tuesday Morning, too. Amazon is obviously very good. You might try poking around at the online "Knife Merchant" shop. He has a great selection and very fair pricing.
A few of the other specialty cooking and professional cooking online sites offer good prices, but I haven't used any nor done enough research to talk about them. Other people here might be able to help; and just Googling and comparing prices (and return policies) should tell you enough.
FWIW, we replaced nearly all of our cookware over the past year buying just about all of the new stuff at SLT. Great service, and their sale prices are the same as everyone else's. I should add that I get a professional discount there.
Hope this helps,