At low levels, jewelcrafting is primarily concerned with the construction of rings and necklaces, which few of the other professions can create. These items can be created and used by yourself, or traded to your teammates, or sold on the auction house - they aren't soulbound, nor do they require the wearer to have any skill points in jewelcrafting to wear them, as many of the engineering items do. Many of them are unique, though, so if you see a specific ring that you like and make yourself two of them, you'll probably only be able to wear one of them at a time.
A Note Of Caution
One thing you should be aware of if you're thinking about taking on Jewelcrafting as a skill for your new character is that training for this skill in Azeroth is only available in the two new capital cities, Exodar and Silvermoon. Characters that aren't Draenei or Blood Elves can still become jewelcrafters, but will have to deal with a fairly lengthy trip to one of these cities before they can pick it up.
The Exodar is pretty, but it¡¯s a long ways away from most of civilization.
You can get Jewelcrafting training in the Outland after level 58, so at that point the problem is less severe, but it's still annoying! If you don't want to deal with a number of long trips that stall your experience-gaining, then you may simply want to use mining as you level up your character and stockpile the ore that you collect on a mule character, using skinning as your secondary profession to sell the leather and make money. If you save six or seven stacks of each type of ore and stone, you should be able to drop skinning and flip over to jewelcrafting just before you hit the Outland and get most of the way to 300 skill without having to go out collecting any more. It's a lot of work for the same result, but it will let you spend your time worrying about getting experience instead of flying back to Exodar every few levels, then it'll probably be the wisest path.
Jewelcrafting is best obtained alongside a new character, although it's obviously available to characters that have already been leveled, assuming that you have the Burning Crusade expansion released. As mentioned, you'll need to use plenty of copper, tin, and bronze to get yourself up past the first 100 skill points or so. You can buy these on the auction house, but since jewelcrafting is the only new profession in Burning Crusade, you can expect that many of these materials will be at an inflated price for a while as everyone and their brother attempts to try out the new tradeskill. Best to farm the materials yourself.
When you start creating items, you'll want to try to increase your skill in jewelcrafting by making plenty of Delicate Copper Wire and Bronze Settings. Delicate Copper Wire is the recipe that should propel you most of the way to the fiftieth skill point, which is where plenty of new recipes will be unlocked to you. When you reach character level 10, you can start creating some very basic rings for yourself. Since rings don't usually start popping up as drops or quest items until the middle or late teen levels, you'll have a bit of a head start on other characters in this category, but the bonuses from the early rings and amulets are going to be very minor, on the order of +1 or +2 to a single attribute.
The Stone Statues, which come along in Rough, Coarse, Heavy, and other stone varieties, will be another great way to level up your skill in jewelcrafting. These statues use up the supply of stones that you will be gaining from mining without drawing down the supply of metal bars or ore that you'll need for your real recipes. They're actually handy to have in combat, as well, since they'll initiate a heal over time when you drop them at your feet; if you aren't a healing class, then these should help you survive some of the tougher fights you encounter as you solo yourself up past the teen levels. They do use up a lot of stone to create, though, so you may want to check the auction house to see if you can't find any cheap auctions for stone. These statues are bind on pickup, however, so they can't be traded or auctioned.