yes? you start with 1 bond slot per character. If you complete a tamer scroll event you will get an additional. There is no limit to how many bond slots a single char can have.
I think most people would consider a comfortable number of bond slots to be 3 total per tamer (that means you just have to acquire or buy 2). That way you eventually have a mare and 2 dragons which is a really fun combination to use for farming. Personally I always bond a mare first, and then expand to 2 dragons however I think many people start with 2 dragons and just use a non-bonded horse (you're a tamer, you can get unlimited horses for free in just a few seconds). But as a tamer, why stop with a horse? Go north of Papua and tame a handful of frenzied ostards and bring them along. Once one of your dragons are tanking the MOB, throw your ostard onto it for a bit more DPS. There are also some MOBs that are really great to farm with non-bonded low-stat dragons (trash dragons). I used to farm exclusively on trash dragons until I got over the *sting* of them dying and realized the time saved farming with trained dragons outweighed the inconvenience of re-training dragons periodically. Not sure you wanted to read all of this, but if I were a newbie tamer I would!