I second the recommendation that cut is the #1 C to focus on.
Color and clarity is in the eye of the beholder. If it looks white and clear TO YOU, that’s all that matters. If you’re looking at an F and an I and they look the same color to you when you have them on, don’t waste the money on the F just to say you have an F. Same with clarity.
When it comes to carats, remember it’s a weight measurement and depending on how a stone is cut it can look bigger or smaller (a not so great stone will be compromised on cut to keep as much weight as possible because it’ll be worth more but it does not mean it will always look bigger. Maybe it was cut deep to keep as much of the rough as possible, so a 4 ct badly cut stone might ‘face’ smaller than a 3.7 ct with perfect proportions. A stone cut too shallow to have a bigger ‘face’ will also compromise the optics of the diamond and how much light it reflects. A GIA triple X or equivalent is the minimum to look for but doesn’t guarantee perfect proportions either.
I would go to a reputable diamond dealer instead of paying the extra for a Tiffany, Cartier, etc. At that range you’ll pay more and the diamond won’t be any more spectacular than what you can get anywhere else. If you were talking about rarer diamonds, then I would say definitely try one of the big diamond names: Graff, etc.
Also, don’t go for the round sizes: 1,2, 3 ct. You’ll get a better deal and won’t be able to tell the difference if you get a .95, 1.90, 2.9 ct instead.
Also, try different style martini settings once you find the size you like because they’re not all created equally. Once a stone is a certain size, if the stone is set too high because it’s in a martini setting made for many sized stones instead of made for your stone size, they can tip over or point one way in one ear and another on the other side.
If you get bigger stones, get jumbo backs and if you want piece of mind, I would go with platinum and the backings you have to push the sides in, then push the back until it locks into place (I forgot the name). They’re a bit more and fussier at first to put on, but worth the piece of mind. I still have jumbo ones on mine and have wanted to change them for a while.
Do not buy a stone without a GIA or I forgot the name of the other reputable institute. Those are the only guarantees you’re actually buying what they’re telling you’re buying.
Not sure whether buying from a place that offers a ‘buy back’ option (or whatever is called) is worth it or not (they buy back what you bought for the same price if you someday decide to ‘upgrade’. It was included when I bought mine.)
This is all I can think of for now hehehe hope it helps!