Cross-class training is necessary to get the upper level jobs (like Scholar and White Mage), plus it helps you net additional abilities for you current class.

Also, if you're playing a healer, you'll want to cross class train as a Thaumaturge as well because they get two abilities that will be invaluable to you later on: "Surecast" (The next spell you cast will not be interrupted *unless* you're moving) and "Swiftcast" (the next spell you cast will be cast instantly).

Also something to be aware of when choosing a Healer class:

White Mages/Conjurers are a bit easier to play, their single heals are more potent and there's less mechanics involved in playing them - however their drawback is that they don't have nearly as good damage mitigation or prevention.

Scholars can actually end up being more powerful healers if they are used at their full potential, but they are quite a bit more complicated to master and there's a lot more mechanics to focus on.