The dragons don't get much focus in the first book; it wasn't until the next book that Anne McCaffrey started actually writing out what they were saying, instead of just describing the gist of it. F'nor's Canth is a bit more of a vocal character in Dragonquest, but Ruth in The White Dragon sets the bar pretty high after that...most books after that generally have a more scenes with dragon dialogue. In addition to that, Lessa and F'lar arguably diminish in importance slightly as the books continue...I mean, they're still authority figures, but they don't get as much time as other characters (and they also get a bit of a personality and relationship rewrite to replace the "Sixties Abusive Romantic Relationship").
From memory, the books with the most vocal dragon characters include:
*Dragonquest - Canth, as mentioned, although oddly other important dragons like Queen dragon Wirenth don't get a single line of dialogue!
*The White Dragon - Ruth, as mentioned.
*The Impression - Golanth...this is a short story included in the "Dragonlover's Guide To Pern" that depicts what Impression's like (the chaos at Ramoth's hatching and Impression was kind of a one-off...call it Early Installment Weirdness).
*Moreta: Dragonlady of Pern - Orlith, Moreta's queen...they're much nicer characters than Lessa and Ramoth.
*All The Weyrs Of Pern - Ruth again, largely because his rider's one of the main characters in this, and also Golanth.
*The Dolphins of Pern - Gadareth...he's not in the book that much, but his scenes are still memorable.
*The Skies of Pern - Lots of dragon characters; but mainly Golanth and Zaranth, who spend several parts of the book chatting with one another!
*Dragonsdawn - Caranth, in later chapters.
*Red Star Rising - Morath; the cute and hilarious green hatchling! Her first real scene always makes me laugh...she wakes her rider by leaning over her bed with lots of apologetic thoughts about being hungry, only to startle her awake and make their heads collide.
Several of Todd's books focus on dragon characters, but I don't think they're as memorable as these ones (certainly, Talenth seems a bit dim...probably cause she's quite young...) and the events of Dragonsblood will just make you sad.