Single-jointed eggbut and loose ring snaffles are your basic bits. A D-ring is popular in the US too. You should only need to move onto anything 'harsher' if there is a specific reason and many horses will need an adaption to these bits to make them even milder in order to be happy.
For starting a young horse, a very nice mild bit is good. Usually a french link (three-piece mouthpiece) snaffle - make sure you know what you are buying and it's NOT a Dr. Bristol mouthpiece!
A full-cheek (eggbut with long bars) or fulmer (loosering version) snaffle is probably the best style of snaffle to go for with a youngster. Whether you choose a fixed or loose ring depends on the horse, but many youngsters prefer an eggbut as the bit is more stable in their mouths. Either of these bits are nice for establishing bend and cannot slide through a horse's mouth.
An alternative to acheive the same function is a hanging snaffle (also called a baucher bit). Contrary to its appearance it does not apply poll pressure.
There are tons of variations, including different materials for the mouthpiece to make it more appealing. There is also something called a 'breaking bit' which is used to gets a horse used to having something in its mouth, but it's not really designed for riding in.
This site has a brilliant section on bits: http://www.sustainabledressage.net/tack/bridle.php