Great advice there by Chipshank, just to reiterate what he said about pick size too - its 100% down to personal perference. Trying to break it down to the width of the pick versus the drag coefficient as the pick passes over a string of a certain gauge just isn't realistic.
Back when I started out I was so broke after getting my first guitar that I couldnt afford the bus fare to get into town to buy a pick so I used my video card and student id (credit card sized piece of plastic. I even used coins when I had... but they were the very last resort!
Now that I can afford the 50 cents for a pick I use hard Dunlop .6mm exclusively:-)
As far as picking fast goes its not about being able to move your right hand up and down that is the problem or being able to move your left hand fast enough, it's getting the two synchronised that's will get you from 50bpm to 200bpm over night.
Here's a good exercice developed by Michael Angelo Battio to determine how fast you can realistically expect to get in a short space of time.
1) Pick an open string up/down/up/down (alternate picking)
2) Start slow then build up over a minute or so to as fast as you can possibly go without losing rhythm or accuracy
This speed is you current 'maximum picking velocity (MPV)' - so now you know how fast you can go all you have to do is get your left hand to move in tandem with your right.
3) IMPORTANT - once you hit your MPV look at how you are holding your pick and picking hand position and make sure that you hold it this way all of the time - even when you play slowly. This will probably change over time but if you are beginning and you want to get from A - Z in the shortest space of time then you should try this for a while. I would say 99% of sloppy players are sloppy becuase they practice playing fast and are 'ok', practice playing slow and are 'ok' but they can't switch from fast to slow in the same piece becuase they hold their picks, guitars and fretting hands in different positions depending on the tempo. When you become an advanced player you will start playing around more with your pick position and picking hand angle etc depending on what you are playing but by that stage you will already have gone from A -Z and back through everything in between.
Bottom line: don't worry about switching picking position untill you have mastered the above.
If you think your left hand can't move fast then try a basic legatto run to see how fast you can get right now (this can get a lot faster with a little practice unlike the 'MPV' which takes a long time to increase).
Anchor your first finger (1), then simply drum your fingers 123123123123123123 etc as fast as you can - you'll realise it's a lot faster than you have ever played before.
Now all you have to do is include this along with the drill and exercises the other guys mentioned (chromatic 3 and 4 note per srting exercises are best when starting becuase you don't have to worry about scale positions etc). Start out extremely slowly and focus 100% on the movement and honing the movement down to something that is comfortable and gradually build up speed. Get this right and you will have a firm grounding to improve on.
Look at the way you hold your guitar and the way you sit. A lot of shredders put the guitar between their legs resting on their left knee with the head stock pointing up somwhere between rib and shoulder height - simply because this takes the pressure off the wrist and is a comfortable way to sit for hours on end while doing mindless speed drills! If you lose the rhythm then don't try to play any faster and ALWAYS play with a metronome or drum machine. Set yourself goals and move the speed up in 5bpm increments - even after 1 week of this at 15-30 mins practice per day you should notice considerable improvements in speed and accuracy.
The above will help you with the raw mechanics of alternate picking - it's not about playing something that sounds 'good' or cool. Give it a go and it will help you to see the potential speeds you can achieve over the next few weeks/months.
Hope that helps.
