What interesting 3-player strategy games are available?
I really, really recommend Ticket to Ride: Nordic Countries as an interesting and balanced three player game. Its unique selling point, over the other Ticket to Ride games, is that it is specifically designed for three! You can play a two-player variant, but you can't for four or more. So you're genuinely getting a game that works best for three.

The other game that is universally acclaimed to be amazing, and is very good for different number of players, is Agricola.  Agricola gives you a different initial board layout for three players, and has many Occupation cards that are specially designed for 3+ players (in addition to the standard ones for 1+ players, and some for 4+ players).  You really feel like the designer paid lots of care and attention to making sure the game would work amazingly for 2 and 3 player games, as well as for larger groups.  (And indeed there's even a serviceable solitaire option!)