All three of them are lovely buildings - the Barberini has famous stairs, as well as a guard cat, but I think it would be the Corsini I would like to live in the most - the views looking up to woods is lovely. However, I presume the garden at the Spada down towards the Tiber must have been rather great, too - but the whole place was much pockier than the other two.
All three had lots of paintings. Lots and lots, crammed all together, mainly of saints wringing their hands/being killed or Magdalenes repenting with bare bosoms. Not to my taste but as I've said before, looking at second rate paintings can be very useful in my own painting - they are much easier to analyse as to what works and what doesn't, and why - for example, I saw two St.Francis's by Caravaggio - one in the Barberini, one at the Capuchin Crypt - and I felt that the Barberini one was preferable - it is less sharp, has more humanity and mystery to it - and was pleased to note that the scientists agree. And this is something I need to remember when painting - lack of definition can be a good thing (his hands, for example, have a clearer junction between the fingers in the copy than in the original.)
If you can only go to one of these places, choose the Barberini. The Corsini was nice, had some lovely stuff including one of Rubens heads, but the Barberini has a nice Caravaggio or two. As well as few more stunners - and I never got to see floor 2, so there may well be even more than I saw.