Does anyone know if simply touching bullets could give you lead poisoning?
Simply touching bullets couldn't give you lead poisoning, since nowadays bullets are usually coated in a protective copper shell.
What makes bullets dangerous isn't the actual bullet but the way lead is expelled from it upon shooting.
When a bullet is fired, it gets so hot that that lead actually vaporizes. ~Sarah Zhang http://www.motherjones.com/blue-marble/2013/01/lead-shooting-ranges-osha
Thus, just touching the outside of a bullet wouldn't be damaging. The only real ways to get lead poisoning would be by inhaling lead in the air - but even that would have to be in large amounts - or by touching lead directly and then putting your finger(s) in your mouth and ingesting it.
See this link for more information.