Just chiming in to post a few recommendations:
1) I saw someone else already mentioned pricescope.com, but I found them incredibly helpful.
2) I also found this website helpful as well:
https://www.diamonds.pro/truth-about/
3) I ended up buying my wife's ring from whiteflash and had a great experience. If I were to do it all again, I would probably check out Costco.
4) I'm far from a diamond expert, but here's what I gleaned from my research about the 4 C's:
Cut - By far the most important. I poorly cut stone won't sparkle and can look smaller than a well cut stone.
Color - Less important for a solitare. If there's nothing to compare it against you don't really see the differences in color. And going back to 'cut', a well cut stone that has a lower graded color may look better than a poorly cut stone that is colorless
Clarity - Hard to tell what a stone actually looks like based on its grading. You really need to see the stone with your own eyes. Grading is done with a 10x loupe (I believe), but do you ever look at your rings that way? If you can't see any inclusions with your naked eye, do you care? Again going back to 'cut', a well cut stone will hide inclusions better than a poorly cut stone. Also, a talented jeweler may be able to hide any defects under a prong in the ring.
Carat - There will be price jumps as you go increase in carat weight, with jumps at certain points (think .9999 to 1.0 carats). Diamond cutters know this, so they may be tempted to leave some material in place to make the cut stone weight over 1 carat, but this would result in a poor cut and not be as pretty as a well cut .99 carat stone. But the 1.0 carat diamond will probably be more expensive.
Regarding a certification (GIA/AGL/etc). It was described to me as getting a certified purebred dog vs. a purebred dog without papers. One will be much more expensive because you have a piece of paper telling you what it is. Getting a certified diamond makes a lot of sense to me if you considered it an investment with the intention to resell at some point, but are you ever going to sell this diamond? That said, I did get a stone with a certification, mainly because it seemed to be the default option from the jeweler.