I think that you could do all 3 areas fairly comfortably in two weeks. The Northwest 4-5 days; Mendoza 3-4 days and BsAs the remainder of your time. But it is a good bit of traveling. I would suggest that you continue to read about the various side trips from Buenos Aires and see what suits you (and any travel companions) best.

The train is not a good intercity option in Argentina; however, the long distance buses are excellent - with the VIP seating, it is more or less like flying business class.

marnie has pretty much covered it all however, I would add - Do NOT skip Salta! If I had to drop any location it would be Mendoza. Salta itself is a nice city but the main attraction for me was to drive the loop around the north west taking in some amazing sights and scenery. It is easy enough to rent car and DIY or, if you prefer, hire a driver guide. Either way you will see some fantastic sights in the 5 or 6 days the route takes.

We visited Mendoza before we went to Salta. If we had gone afterward it would have been a disappointment. Nothing wrong with the place at all but the main reason to go there seemed to be wine.

The "best/least expensive way to travel between cities" is by bus. Air travel is horrendously expensive for foreigners and not always as quick as you may think as delays are quite frequent. There is no inter city rail travel in Argentina although there are some good shorter trips, notably from BA out to Tigre and from Salta to San Antonio de las Cobres on La tren de las nubes - http://www.telegraph.co.uk/travel/journeysbyrail/4581018/Salta-Argentina-Train-to-the-Clouds.html

Travelling by bus is around 25 to 30% of the cost of air travel and does not eat into your holiday time as most long journeys are scheduled overnight ( saving on accommodation costs too!) choosing Ejecutivo or firsts class will provide lie flat beds and comfort similar to first class on a good airline. We used this form of travel all over Argentina.

It is now 4 years old but there are some photos of these places and some more info on our blog. Entries #73 on http://blog.travelpod.com/travel-blog/candcthai/1/tpod.html

Great info, Crellston; I certainly agree about the Northwest being spectacular. Iguazu Falls could be added for 2 days with a flight from Salta to Iguazu (or vis versa) without going back to BsAs (flight available only a few days per week). Unless the OP has a particular reason for going to Mendoza, I would consider Iguazu Falls and the NOA higher on the favorites list.

Train to Salta? I am sorry, but no trains coming to Salta form Mendoza. There is one big risk for you to come to SALTA in April; that you probably fall in love with the NW and move!!!
Do not miss to taste the torrontes wine!!!!
Goold Luck!!!