I bought a diamond from Blue Nile a long time ago. I did a bunch of research ahead of time to make sure I knew what I was getting. It went very smoothly.
2 + 2 = 5 (for extremely large values of 2)
You are not a snowflake. "It is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing."
Online? Na, not for me. Try this. Fly with your GF to Thailand (Chantabury). Buy some insanely big and clear stones from street brokers. Chantabury still has some intresting stones, like sapphires an rubies, star rubies etc (no diamonds though). You can talk to same dealer about making it into a ring, get measurements, review design etc. Then have a nice vacation, couple weeks on a beach, great Thai food and silky smooth beaches. Pick up your jewelery and go home with good memories and great rings set. While being in Chantabury you can also load up on some stones you don't really need, and re-sell them back home for at least 5-fold. That is if you shop carefully and sell carefully. Sounds complicated but it is not, and your trip is paid itself off with some "leftovers" to show for it.
Location: Guildford, Surrey, United Kingdom (Great Britain)
Stats: 5'11", 178 lbs
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If you want serious diamonds, then by far your best bet IMHO is to go to one of the many "fine *********" auctions by the big auction houses, for example Bonhams. For a given budget you will get three times the bang for your buck or more compared with buying new, whether on-line or high street. Plus you can (and should) buy one which comes with full certificates of authenticity etc for insurance purposes (and god forbid, resale should it ever be necessary!)
The auction process is a bit daunting (you are normally competing against diamond dealers), but my experience was that they recognise engagement ring punters immediately and back off when you pile into the bidding - they know their margins are much tighter than yours are likely to be! That presupposes you are bidding in person - but they all do phone bidding - the only difficulty then is getting to look at the ring/stone in question. But even then you will know as much as you will online.
Diamonds are a bit like something else. Women claim that size doesn't matter. But they are lying . If you can get a stone three/four times the size of something from Blue Nile, but of the same quality and at the same price, then why wouldn't you? Likewise if you want something small if you can get three/four times the quality, or at 1/3 - 1/4 the price, why wouldn't you? The actual setting doesn't matter much - getting a stone reset is very cheap compared with the cost of the carats...
Just to give you an example of the sort of pricing a 2.67 carat old brilliant VVS1 G was just under £10,000 a few years back. I just checked the only comparable stone on Blue Nile today and it is £40,000.
Obviously not suggesting that your budget is going to be that (it may be much bigger ), but the differential is similar across the board at least until you get to very big rare stones which sell at a premium even at auction.
The only conceivable downside is if you (or your intended) feel funny about something being "second hand". If so, see if you feel any better with "antique"!
Alternatively as Weiss says you can fly somewhere nice to find cheap stones. But I would not begin to have the confidence in my knowledge or skills to be spending big money on stones abroad.