Welcome to FFXI! You're absolutely right that a good LS is the key to whether you have fun overall in this game vs. spend long hours struggling. (Well, MORE hours than you otherwise would.) It's a Japanese game culturally, and IMHO that means lots more need for teamwork, although changes in recent years have lessened the need for parties and teams somewhat.
I suggest you do two things. Generally, check out the stickied thread atop this (Ragnarok server) forum for linkshells. Some will be defunct but many are still kicking. Shop around! Not every linkshell that's good is a good "fit" for everyone. It's about personalities clicking.
More specifically, you can check out the four linkshells where I spend my time. My primary home is Axiom, where I'm one of the leaders (axiom-online.org/forums). The other three are Musketeers, Triumvirate and MogHat, all friendly places, each with its own mix and flavor.
In-game, I am Grommet, but I can be hard to find on weekdays because I only get on for about 45 minutes per evening. Weekends I usually get a bit more time.
____________________________Grommet Taru male San d'Orian
95 WHM, BLM, RNG, SMN, WAR/ 64SAM/ 52DRK/ 51PLD, THF/ 49NIN, SCH, RDM, DNC
100 Woodworking and Fishing; 80 Synergy
60 in all other crafts
(Updated November 12, 2011)