There's a few ways to do this. If you just want to quickly check snapshot of prices without regard for the economy of your particular server, just check Wowhead.
Wowhead displays the items vendor value, as well as the average auction buyout price across all servers.
However, this information is sub-par in that not every servers economy is the same, and in fact substantial price swings occur across realms. Also, for items that are less frequently traded, there may be little to no data.
If you want a more detailed picture, you need to go to the source from which WoWhead derives it's average; The Undermine Journal by contrast, provides a much more comprehensive picture, giving you price history and current pricing data for any item, specific to your realm and faction, as well as the cost of component materials for crafted items, should you wish to just produce them yourself or ask a friend to. Additionally, TUJ distributes an add-on that will allow you to view historic data for an item in game while browsing the AH.
As you can see from this screenshot, The Undermine Journal is far more detailed, as well as providing a more specific and accurate picture - for instance, the item in question (Flask of the Winds again), is roughly 30% more expensive at the Ysera-Alliance auction house than Wowheads average across realms indicated.
(European players will note that TUJ EU site is found here.)
Carnelians are definitely worth keeping, as you can transmute them into even more Inferno Rubies.
Zephyrite, Jasper, and Nightstone are all used for dailies, so list these on the AH when the appropriate daily is up.
The best use for Alicite and Hessonite, depending on AH prices, is to craft Alicite Pendants and Hessonite Bands for disenchanting.
Alternatively, depending on prices, saving up all your green gems and using them to transmute to meta gems can also be some good money.
For blue quality gems you're still stuck figuring out which cut will make you the most money and posting them on the AH appropriately. There's not really a better way than that.
Best Answer
There are no "Sell Orders" in World of Warcraft, however you can put your own item up for auction by using the "Auction" tab with the auction house window open.
If you want other players to buy out your item right away, set a "buyout" price. The "starting" price is mandatory, but can be set to the same value as the buyout price to discourage bidding.