The bullets are lead-free, but usually the primers still contain lead which is a big, maybe the biggest, part of lead exposure with firearms, and it can be pretty hard to find ammo with lead-free primers, it also can get pricey, and by a lot of accounts they're less reliable than lead-based primers, so that's something to be taken into account.

Lead exposure from the ammo itself is usually pretty minimal, a lot of bullets are jacketed in copper so under normal handling you're not really going to come in contact with much if any lead. There will probably be some fouling in the barrel, but that's mostly contained in the barrel and unless you're cleaning your gun on the same towel you use to wash your face thats pretty easily contained with some basic precautions like wearing gloves and such (not that all, honestly probably not most gun owners take those precautions seriously)

But when the primer detonates it's putting out a lot of aerosolized lead compounds that kind of get all over the place because it's basically getting sprayed around you- on your hands, on your clothes, the outside of your gun, your range bag, you're inhaling it, etc.

So keeping your gun and shooting gear in a safe or something helps to keep that contained so it's not getting transferred from your gun and range bag to wherever you set it down.