Usually ingrown nails (fingers & toes) come from clipping one too short, which is so hard to avoid on an infant who is moving and whose nails are so tiny. I agree with Tadpole about the mittens or those long sleeved shirts with the fold-over mitts.
One side of my family has a hereditary propensity for ingrown toenails, which relate to the way the nail originates at the matrix. Basically, they grow too wide for the toe and often need to be worked on by the podiatrist. But if it's a problem more at the edge of the nail (where you clip), the traditional home remedy is to push a tiny piece of thin toilet paper under the edge that's trapped under the skin. We're talking the tiniest corner of a single ply sheet. Use the thin pointy attachment on many standard nail clippers or perhaps the end of infant nail scissors if flat enough. If a nail is super red and inflamed, this can hurt but you do it anyway. You leave it in until the nail grows out enough - which in your case shouldn't take long. But your baby seems not to be in pain you said, so it's probably very easy to do. Your probably want to put a mitten on that hand, although the piece of TP is so small it's probably not a risk even to a newborn. We don't usually use neosporin - it's not usually necessary. The redness can be from irritation rather than just infection, so I'm not sure the neosporin serves any purpose if the skin isn't broken.
