My Chocobo was Rank 7 when Chocobo Stables were released and now it's 17. All I've been doing is feeding it Krakka Roots from my garden for a few hours every day. The trick is that your chocobo can be fed twice every hour. Once by your own character, and once by someone else in your FC (my girlfriend does this for me, and I for hers.) Other then that, make sure to do your 100 Chocobo Kills challenge once a week for a free 15% of its exp bar. Add it all up and you get about one rank a week this way.

Note: This requires a constant planting of at LEAST four Krakka Root Seeds using Grade 3 Shroud Topsoil, re-planted every time they are ready, fertilized non-stop, to keep up with the Krakka Root demand that two characters will need. You may need more depending how active a player you are but the 72 Krakka Root yield just barely covers what we use.