My husband and I are looking for an inexpensive holiday in March. I've seen great fares to Bucharest. In doing a little more research, it seems that we would enjoy the small towns more, such as Brasov and Sighisoara. Would it be nuts in March to try and train or drive between these on a long weekend? Maybe just to Brasov by train right off the plane (is that doable from the airport?)? Looking for any advice!

I think it all depends on when exactly in March you are going. With luck, there will be no more snow as you head towards Brasov. Even Bucharest has a couple of things to offer so I would recommend at least 2 days since you're flying there anyway. To go to Brasov, you would need to go into the city, take a train from Gara de Nord station. Via the IC train it would take you 3 hours. I advise against it ne reason being a possible stay in Bucharest, the second being that you would not want to travel on your first day in a foreign country like Romania (which might take some getting used to).
Have you looked at other posts on the topic? There are plenty of them that deal with itineraries around the cities you want to visit in Transylvania. Look for Clifton's, Traian's on Bucharest, etc.

I apologize for the 'face' it was meant to be "the".

Brasov is a 150 minute train ride from Bucharest, or two hours from Ploiesti, which is a 15 taxi ride from Otopeni, the main Bucharest airport. There's no logic in when they go - but there are a dozen or so a day. TRy www.cfr.ro (there's an English version somewhere on the site)

If you're familiar with Romanian trains, going straight to Ploiesti from Otopeni saves you 45 mins or so on your journey.

IMO

I would do a Budapest weekend break or Vienna rather than Bucharest in March. We are talking 2nd world with a verneer of 1st world over it and the trains are interesting