Buying diamonds online is probably less risky than buying them in a jewelry store. I've seen lots of online places that sell fake diamonds (moissanite and CZ), lab-created diamonds, and diamonds that have been color enhanced or clarity enhanced. However, they are all clearly labeled as such.
I prefer do to business with people who don't carry that kind of merchandise in the first place. Online places I would recommend are B2Cjewels (10% above wholesale), Blue Nile (20% above wholesale), James Allen, WhiteFlash, and Brian Gavin Diamonds (all a bit more expensive). Plus a few bargain places: Union Diamond, Brilliance, and Enchanted Diamonds.
If you get a diamond with a certificate, it should be a first-class certificate. That would be GIA, AGS, or HRD-Antwerp. The last one isn't available in the US, and AGS is kind of rare, so that pretty much narrows it down to GIA.
At Blue Nile, 99% of their diamonds are GIA-certified, the other 1% AGS-certified. B2C, Union Diamond, and Enchanted Diamonds are a mixed bag: they sell junk diamonds with junk certificate right alongside GIA-certified good diamonds. (If the price looks too good to be true, it's usually one of the junk diamonds.)
A GIA certificate doesn't guarantee that it's good diamond, just that it's a real, natural, untreated diamond that has whatever qualities it says on the certificate.
I've been trying to find something in your budget, but it's very time-consuming. Blue Nile had a pair of platinum settings for studs for $250, but their cheapest pair of diamonds (0.46 carats) was about $600 and not very good quality. For $50 to $100 more they had diamonds that were halfway decent.
The reason platinum is a good choice is because white goes better with diamonds, but white gold is actually rhodium-plated gray gold. It tarnishes and the plating wears off. Platinum is more expensive because it only comes in high purity (900 or 950). 14k yellow gold is the next best choice for people on a budget.
It's a good idea to stay away from diamonds with a clarity rating below SI2, and preferably below SI1. For color, J is about as low as you can push it, I is better but not great.
Bezel is just a style of setting where the metal overlaps the edge of the diamond. Most diamond experts say it's a terrible thing to do to a diamond, since diamonds need light.
If you mean the shape of the diamond, round brilliant is the most popular shape, accounting for 50% of all diamond sales. Princess cut (square) is the second most popular, accounting for 25% of diamond sales. Round brilliant has the most sparkle, but is the most expensive, and to some people, the most boring-looking because it's so common. At one carat round brilliant is about 25% more expensive than princess cut, but at 1/2 carat the difference is less.
By the way, the website you mention sells IGL-certified diamonds. This is not a reliable certificate. On the other hand, it's nice that they tell you what the clarity-grade is and let you pick which one you want.