When you choose the right tool for the right job and learn how to use it the right way, it all works out fine
hey Chris I think G.G. is making the point here. No one type of cookware is all purpose (if they claim to be it is marketing not reality) different pans do different jobs better. This is why buying a variety of cookware rather than a set will ultimately serve you best. Any set of all one type will end up disappointing at one task or another.
I have a lot of cookware... ss, copper, enameled cast iron, raw cast iron, glass, stoneware. All of it at least reasonable quality some of it very good. 2 things I did not do were spend a lot of money or buy it all at once. I bought as needed to make things I wanted to make and always looked for deals.
From your list
Steak and hamburgers and grilled chicken are a natural for a Cast Iron skillet or grill pan... non stick or even SS or anodized will not do this as well. Get your skillet well seasoned and it will require virtually no cleaning after a quick grill
Spaghetti... pretty much any stock pot can boil water your choice here... but for making rich thick spaghetti sauce you want either a enameled Cast Iron or a Heavy SS saucepan that will allow you to heat evenly for long periods of time and not react with the acidity of tomatoes. For a quick fresh tomato sauce a SS saute pan will work great.
Rice... I use a rice cooker
because It is one of the things I tend to ruin but a good quality saucepan will work... here you might want to go with non-stick.
Gravy if you are making your own you want a bit of sticking so you can deglaze the fond this is what makes it tasty... a good SS or anodized will work as will a well seasoned raw or enameled CI skillet. Non stick not the best for gravy
Teriaki chicken... my choice would be enameled-CI or SS and soak the thing but this may be another place where you choose Non Stick