Years ago, I purchased a set of Member's Mark (I know you said no sets, but they are often less expensive than buying individual pieces) from Sam's Club. The set is made by Tramontina and all the pieces are three-ply construction with a mirror polish. They are absolutely wonderful. I don't know if the set is still available. Costco has an impressive set under their Kirkland label. The pieces are very heavy and look like they would do a good job, but I can't vouch for them. Neither set has any non-stick coating.
I have a ceramic cooktop and have had no luck with any aluminum pot/pan, even from the expensive brands. They warp and spin around on the burners. You didn't say what kind of cooktop you have, so you might want to consider what material would be best.

The only cookware materials that I'm aware of that even have a sniff of health-related controversy attached to them are bare aluminum, and teflon-type non-stick. Understand that these materials are "controversial" but that there are absolutely no scientific studies linking them to any adverse health conditions.

Now that that's out of the way, I wouldn't recommend just one material for cookware, because there are many materials used, and some are better than others for specific cooking tasks. My favourite pieces are plain old cheapo cast iron. Actually, vintage tends to be better than much of the brand new stuff, as they cast and polished things up better half a century ago. Even still, these are cheap compared to primo ss brands. Cast iron, when properly seasoned, is non-stick. Because of its mass, it holds heat very well for searing meat. It goes from stovetop to oven, and fry pans can do double duty for baking cakes. Healthwise, it is often found that folks who cook with cast iron are less likely to suffer iron deficiency.

I like stainless steel for saucepans, when you are cooking with a lot of liquid - boiling/steaming veg, heating soup, etc. Also for stock pots. Stainless pots need some type of "cladding" or encapsulated plate of a higher conductivity metal (aluminum or copper) because ss itself is a relatively poor heat conductor. Pots with the cladding up the sides are more expensive than those with just the extra conductor in the base.

An enameled cast iron dutch oven is great for braising or stewing. The enameling ensures that the cast iron doesn't react with acidic ingredients, which can sometimes impart funny tastes to the food.

While it is fairly easy to avoid teflon and/or aluminum in one's own kitchen if you want, if you want to be rigourous about it, you will never again eat out at a restaurant. Most eateries I patronize use aluminum pots almost exclusively. They conduct heat very well, and they are cheap. Restaurants wear out and replace pots and pans wayyyy more frequently than most folks do in their kitchens, so price does matter. Frankly I appreciate the multi-generational aspect of my cast iron.

What I have done is go with Enameled Cast Iron (LeCruset) type cookware for pots.
Cast iron and Carbon Steel for pans.

All of these are true lifetime cookware, no cancer causing agents of any kind, get better with age and cost less.

Buy yourself a 12" cast iron frying pan like a Lodge under $25.
Then either a 14 1/4" OR 15 3/4" carbon steel "Lyonnaise" pan $63-$75 these pans are every bit as good as a cast iron pan with a bit less weight and once seasoned 100% nonstick FOREVER, pass them down for generations.
Could also pick up a 9 1/2" Debuyer pan if you feel you need one smaller frying pan. Cost is $33.

http://www.bakertowne.com/1620/de-Buyer-511040--15-3/4%22--Heavy-Duty-Frying-Pan-lyonnaise.html

Then your final "pan" should really be a carbon steel griddle, if you use a regular size range then buy a 12x20" if you use a professional size range then buy a 14x23".
Buy a Chef King carbon steel griddle, cost around $70ish.

http://www.amazon.com/Chef-King-Gauge-Steel-Griddle/dp/B001BQYS3C

Then your pots I would buy LeCruset/Staub, maybe even the Lodge enameled cast iron or if you get lucky a Kirkland/Costco brand what ever you can find on sale and what size you need.
Myself I went with a 3 3/4QT LeCruset multipot which is about 10 1/2" in dia and the lid is a nice frying pan. Cost was $200.
http://cookware.lecreuset.com/cookware/product_3-3%2F4-qt.-Multi-function-Pan_10151_-1_20002_41069_10032

I am going to pick up soon a 2 1/2QT LeCruset multipot which is 8" in dia and the smallest pot that I feel is usable on my range. Cost is $170.00

http://www.surlatable.com/product/PRO-1035534/Le-Creuset-Cherry-Two-in-One-Pan

I luckily found an 8Qt Oval enameled cast iron roaster from Costco for $79.00. The LeCruset version is over $400.
So do what you can on this piece as cost can be between $80 to $400+.

I would also get a 7-8Qt stock pot/ Pasta pot to use an an all around Stock/Soup/Pasta/Steamer etc type pot.
This is the ONLY pot that I would buy Stainless.

They are a pain in the butt to clean but they work well, boil water quickly, last, etc
The Pasta pot is more versatile with the pasta strainer insert and then buy a steamer insert, you can use it for anything you need a large pot for. The good one will cost around $300 then another $100 for the steamer insert.

http://www.amazon.com/All-Clad-Stainless-Pentola-Insert/dp/B004T6JAAC/ref=sr_1_5?s=home-garden&ie=UTF8&qid=1351961150&sr=1-5&keywords=pasta+pot+8+qt+all+clad

This "set" will cost somewhere in the neighborhood of $1300.00 but the pans are totally nonstick and get better with age, last several lifetimes.
The pots are virtually nonstick and should if well cared for last at least your lifetime.
They also cook BETTER than any other type of cookware.