What could all the tiny bumps over my babies arms and legs be?

Possibly ringworm. Ringworm is very common in kids - by some estimates, up to 1 in 5 have a patch at any given time. The classic ringworm rash starts as a dry, slightly raised, mostly round patch, which may be slightly red. The patch may also be somewhat itchy. As it grows, the center of the patch often clears, leaving a ring-like appearance...More info: http://www. Drgreene. Com/azguide/ringworm.
Lots of things. Little bumps could represent many things. A reaction to: a detergent used to clean the clothes s/he wears; to a dryer sheet or fabric softener; to a lotion applied to those areas; a final stage in a viral illness coming to the skin. Skin bumps are often diagnosed after a visual inspection, but not always. If it seems to bother the kid it may be worth a visit.