Bearded dragons like any animal can bite. Young dragons are more likely to bluff and try to bite but typically with patience they will learn there is no need to bite. Once comfortable with their 'people' bearded dragons are calm and quite happy to be held, sit on a lap or shoulder etc.
The spines are not as dangerous or sharp as they look but can prick a finger or hand if aligned right. I have had a few little pricks on my hands when picking up my dragon but none were painful or came even close to breaking the skin. Maybe a little dimple for a couple seconds but thats it.
Children should never be left unsupervised around dragons and hands should be washed before and after handling dragons. If handled gently most dragons are calm and tolerant of being petted and held. Keep in mind that not all dragons as as quickly tamed as others and some remain rather resentful of being picked up most of their lives.
Dragon claws can scratch hard enough to break the skin if they are struggling to get away. If they are calm there is no worry about being scratched. Young dragons don't bite as hard as older dragons but either can break the skin with their teeth if they get you the right way. Their bites however are not as bad or painful from overly playful cats etc. With dragons they typically don't bite without puffing out their beard and hisssing and gaping first. Some will do mock attacks first but sometimes they skip that part if they feel more threatened than normal.
The best way to pick up a dragon is by coming in from below (preferribly when the dragon itself is at waist level, harder to do this when they are on the floor). Gently place your fingers underneath the dragon till you can scoop him up safely onto your hand, then turn him to face away from your body with the tail between your arm and body. This makes it easier to control the tail should the dragon decide to struggle.