I'm not very good on Buenos Aires specifically but can give you info on Argentina in general as Dh is Argentinian and we go there every year.
Journey Latin America for flights - Lufthansa generally the cheapest. Book well in advance because a lot of airlines have cut back their flights to Argentina so the cheap tickets go very quickly. March isn't peak season for either country though so you should be fine but you really ought to think about booking soon. British Airways is the only airline that flies direct and even they stop off in Sao Paolo for an hour or so. Otherwise you have to go via Frankfurt, Milan, Madrid etc and probably stop off in Sao Paolo too. Lufthansa are very good with children but there are normally lots on the flights so make sure you request bulkhead seats as soon as the baby is born. They probably won't be able to confirm you've got the seats until you check in so make sure you get to the airport very early. Also you won't be able to book a ticket for the baby until he/she is born (we're flying out at end Nov but can't book ticket for the baby until she arrives in early Oct).
We go straight through Buenos Aires and have done since we have had children so I can't really recommend hotels or things to do with kids.
Food is easy for children - lots of pizza, pasta, ice cream, fruit, sandwiches, empanadas (little pies with different fillings, eg ham and cheese, meat, fish), fish, lots and lots of beef, chips, tomatoes. Vegetables not eaten so much but salad with everything. Tap water is fine to drink everywhere.
People take their kids everywhere with them and they are welcome at all hours in restaurants. Most people love children and they will get lots of attention.
March is the end of the summer but it will still be very hot and pretty humid. Mosquitoes around too.
No vaccinations needed, or malaria pills.
Let me know if there's anything else you would like to know and I'll see if I can help.