Getting started with trick-taking games (whist, euchre, bridge, pinochle, etc.)
Spades.  

I think Spades is the easiest starting point, because is (1) easy and (2) the point is winning tricks, making it more consistent with the logic of other games. IMHO, spades shares more in common with the other games.

Hearts has the reverse logic -- you are generally trying to LOSE tricks, not win them.  If you start with spades, you spend your time thinking about how to win tricks.  That means you can switch to Eucre or Bridge or Pitch or whatever...  Or you can learn hearts next.  


Spades to Bridge --> play is the same; add "no trump" and dummy hand; bidding more complicated 
Spades to Euchr --> drop a bunch of cards from the deck, etc... but still  win tricks 
Spades to Pitch -->  multiple versions exist; typically a varient of spades  
Spades to Hearts --> lose tricks, not win them
Hearts to Bridge --> more "moving parts"  than "Spades to Bridge"


In any case, once you know one trick taking game, it's somewhat trivial to learn the rules and basic play of one more.

I'd recommend you avoid Bridge as a first game. The bidding is really complicated. You can start having fun and start feeling competent much faster.  Pinochle is more complicated.... 

However, if you're only going to actually learn one game, I have to say that I find Hearts and Spades to be the most boring.   What makes a game better for beginners makes it worse for experienced players.