Mandolin picks are NOT the same as guitar picks. They are very stiff – too stiff to even use on a guitar. This stiffness is required because a lot of mandolin music – especially classical and bluegrass – is played very fast. If you try to use a pick that is not stiff enough, you will not be able to play fast because your pick will still be flexing back and forth when you need to play a note. If you use a pick that is too soft, you will sound like you have no sense of rhythm, and you've completely missed half of the notes you are trying to play.
Some players use antique picks made of tortoise shell. It is illegal for manufacturers to use tortoise shell for this purpose now, but many of the old ones are still in use because they're tough, and because they perform so well. The players who own them guard them more ferociously than their wallets.
New players have to rely on new technology, which has produced much harder plastics that hold up much better than the old ones did. One manufacturer has even produced a pick made of Kevlar, which is the material used in bullet-proof vests.
Picks come in all kinds of different sizes, colors, and shapes. A few of these are shown below:
In my opinion, you should try every kind of pick you can get your hands on. They're inexpensive, and, depending on the size/shape/thickness/material of the pick, will produce a completely different tone from the same instrument. If I'm fortunate enough while I'm traveling, to pass a store that sells mandolin accessories, I'll stop in and buy a bunch of different picks to try. Some of them I won't like, and some of them I like so much that I regret not having bought more of them.
How you hold the pick is a matter of preference and functionality – meaning that if you can hold the pick in a way that allows you to produce a good sound and do it with no pain, there's no reason why you can't use that position. I prefer to hold my pick by first curling all of my fingers in so that they almost touch the palm of my hand. This forms a "support column" of fingers, and then I set the pick on my index finger, at the first joint, with the point facing straight out. The thumb is then placed over the pick and held by the thumb joint, not the tip.
Notice that this position forces the wrist to arch slightly. I began to use this position because I had developed the bad habit of allowing my wrist to collapse and rest on the strings while I was playing, resulting in a muffled sound. In addition, I used to hold he pick steady by using the tip of my thumb, which forced my knuckle to bend and stick out. Eventually the joint at the base of my thumb, and the joint connecting my hand to my wrist, began to ache all the time. My chiropractor explained that action of picking the mandolin while my knuckle was at that angle put a lot of unnecessary stress on the joints of my thumb. By flattening out my thumb, not only do I get better control of the pick, I don't have any more pain!
One position I recommend you don't use, is holding the pick between your thumb and index finger, with the other three fingers extended out.
Although it's a commonly used position, I've actually seen players who use it, lose their picks completely, and while in the middle of a performance. So I think there's a question as to whether you can continue to control the pick in that position, as you develop more strength as a player.