Consider a vertical rock face with clip-ins 3 meters apart, dynamic rope, no more than 15 meters ascent.
I guess most probable and dangerous is to fall just as one is clipping in. This way the rope is overextended, plus one i holding with only one hand.
How dangerous is it for a for a beginner climber to fall? What protection equipment should be used. What training exercises should be completed beforehand?
How dangerous is it to fall 3 meters when lead climbing?
This depends a lot on the fall factor. Counterintuitively, the fall factor is higher, indicating a more dangerous fall, when you are near the beginning of a pitch. This is because there's less rope out, so there's less stretchiness.
As Steed's comment points out, you're going to fall more than 6 m in this situation, not 3 m. After you get your first clip in, you may have a false sense of security; by the time you're reaching for the second clip (which is 6 m up in the situation you've described), you have enough rope out so that if you miss the clip and fall, you're going to hit the deck.
Different falls can have different run-outs. A fall from an overhang is actually the safest, since you can't hit anything. Depending on the route, you could swing like a pendulum, or slide down a slab and get cheese-grater injuries.
What protection equipment should be used.
A helmet. If you're trad climbing, that's a whole different thing -- we'd be talking about a trad rack.
What training exercises should be completed beforehand?
Start by learning to give a lead belay, and practice doing that enough times so that you understand the whole process well. Learn by watching your climber while belaying. Discuss things verbally with your climber. Help your climber catch mistakes.
Learn not to back-clip, z-clip, put your thumb through the biner, or back-step the rope. Spend some time at home practicing clipping efficiently in a variety of positions with a quickdraw hanging on a doorknob.
If you do fall, absorb the impact against the wall with your legs, and don't grab the rope.
Learn the steps involved when you get to the top of a sport climb. This is actually pretty complicated.
There are some advantages to learning to lead in a gym. For example, the routes tend to be constructed so that the run-outs are very safe, there is no danger from rock-fall, and you don't have to worry about the quality of your anchors.
I wanted to mention two additional scenarios not covered by Ben Crowell's great answer.
When it is possible to fall past the belayer (on a multi-pitch climb, or when getting to the start of a climb involves a scramble or stepping out onto the face,) the fall factor is the greatest. In those cases the old mantra "the leader must not fall" still very much applies. If you fall past your belayer without having clipped any protection yet, you are experiencing a factor 2 fall (falling twice the distance of the length of rope out), which can create enough force to break climbing equipment, rip bolts or protection out of the wall, and cause serious injuries or death to you and your climbing partner, even if you fall for just a very short distance.
(To prevent this scenario, place a piece of protection as soon as possible. Until then, don't fall!)
When you are clipping to gear that is above your tie-in-point, you can fall even further than twice the distance to your last piece.
Example:
You clipped the first bolt, which was 15 feet (4.6 m) above ground, now you are standing 4 feet (1.2 m) above the first bolt and within reach of the second bolt, which was only 10 feet (3 m) past the first. Clipping involves lifting the rope 4 feet (1.2m) from your tie-in-point. At first thought you should be fine, right? You are standing just four feet (1.2 m) above your last piece, and the first clip was after 15 feet (4.6m.) In fact this is a very dangerous situation. When you are just about to clip, the actual amount of rope out is: 15 feet to the first bolt, 10 feet to the next bolt, and 4 feet back down to your tie-in-point, totaling 29 feet of rope, plus maybe a foot of rope of slack your belayer is giving you plus rope stretch that will occur during your fall. In this scenario you could fall 14-19 feet (4.2-5.8 m) past your last piece which is going to result in a ground fall !!!
(To prevent this scenario, avoid clipping to something that is higher than your shoulders, and, whenever possible, clip when your protection is near or past your tie-in-point.)
Climbing is full of little surprises like this (force multiplications, the physics of pendulum falls, the zipper effect, factor 2 falls... to just name a few.) I am all for figuring stuff out myself, but with climbing I highly recommend learning from a climber with several years of experience, or taking a class. Just reading a book or post on a Q&A site is not going to be enough to stay safe until you have a very thorough understanding of everything involved.
