Speaking from experience with taming and breeding a host of dragons, here are some things you might want to look out for.
As above posters have already mentioned, there is a unresolved bug involving military and visitors attacking your dragons(see: patrick lundell's response). The workaround to this is to keep your dragons away from visitors, military personnel at all times. You'll want to house their nests somewhere out of the way but fairly easily accessible. Reason being that it makes it less likely for military squad members to have a reason to visit their nests areas and risk your carefully cultivated dragon stock going bust, and second fold to make them easier for animal trainers to access. Doors and corridors to break line of sight might be useful too if they are situated near a heavy traffic zone.
Also be sure to turn off food hauling, animal trainer and just to be extra safe animal caretaker on your military dwarfs, so they won't have any reason to visit the dragon brooding area. Maybe add restricted traffic zones as a precaution. You can lock them away behind locked doors but be wary that if you don't pay attention to their training needs they will go feral.
Would recommend chaining ALL dragons at all times. If you don't lock them away, newborn dragons are very likely to go wandering all over the fortress. This can ensue in !!FUN!!. I've used chains and nestboxes to great success and this has not disrupted their training or breeding. Cages are another option but they will not breed as they need access to a nestbox. Turn off or disallow dragon eggs in your food stockpiles so you don't disturb them, or you can manually forbid the eggs (not the nestbox) as well. I think the hatching period is around 6-9 months for the eggs? Its been a long time for me, you can "t" over the eggs to check if they are fertile or infertile i think. Not sure if this requires any third party tools. Just remove the eggs from nestbox and wait for a new batch to be laid if infertile. If infertility persists, you may have a problem with your breeding pair. If you have DFHack, you can use gui/gm-editor, mouse over the dragon and go to: status->current_soul->personality->orientation_flags to change them into the correct orientation.
Dragons are expensive pets and so they take a long time to train. You can never fully tame them so they will require periodic training every now and then. However they do give lots of animal trainer exp. Do be aware that they take a fairly long time per training session, so I would recommend turning off all hauling and non-essential skills for your animal trainers. If you have only a breeding pair of dragons, one full-time trainer should be sufficient. If you want an army of dragons, get a supporting army of animal trainers.
Young dragons (and dragons in general) aren't really durable, they are more of a glass cannon, so they don't make great war animals/ military fodder. However, they make great stationary turrets provided you can find a good arrangement where they won't be shot back by enemy archers. Dragonfire only shoots at the same z-level and melts pretty much everything (a few versions ago, steel was the go to material for resisting dragonfire, this is no longer the case). Cave dragons are far more sturdy and make great war animals to bring into combat.