Excellent question, I had to tackle this right out of undergrad as my son was born about a year after I graduated and I was working for the University & living in C-ville at the time. I think in terms of proximity to Darden & not having to worry about elementary schools (they tend to be better in the city as opposed to the county), I would look at places in the Barracks corridor near the Barracks Road Shopping center. I've seen places both on craigslist & BRAC (Blue Bridge Apartment Council) listed for pretty decent rent given the proximity & in fact lived at the Barracks Ct. apartment/condo complex with my wife right after undergrad. Very quiet, very close & good prices with easy access to the rest of C-ville, though activities on the Corner may be a bit of a hike from a public transportation standpoint. And Darden is about a 5 minute drive or 10 minute bike ride. If not there, I understand the most popular Darden neighborhood is off of Ivy Road or the Huntington Village apartment/condos, though those spots seem a little pricier relative to the Barracks area. I'm looking at single family homes in the Preston Ave./Rugby area since the elementary school there is the best & most diverse in the city. A bit further of a hike to Darden, but close enough to the bus to be a livable trade-off for me.
When we were living there we used a diaper service, and I highly recommend it once the baby comes:
https://www.thestork.biz/They were always on time with deliveries and drop-off was very easy. Also, the cost is approximate what you'll spend on diapers for a newborn/infant, or at least it was when we were there. With a new baby, I would absolutely suggest having a W/D or buying one if there are hook-ups. You do an unbelievable amount of laundry & having to haul it to a separate facility with in an apartment complex will get real tiresome very quickly, not to mention how mildly infuriating it is to make sure you always have quarters laying around.
Oh, and the winters in Cville can be cold but they are about half as long as in New England, or at least Dartmouth. Due to its proximity to the Blue Ridge, it does get snow but rarely is it a New England-caliber amount. Summers in Cville can be quite hot. Fall & Spring are absolutely lovely & temperate and the best time to be there in my opinion.