I'm planning to take Dragon Disciple with my bard, and it's unclear whether I would gain the Draconic bloodline's bonus spells (like mage armor at Dragon Disciple 3, for example) through the Blood of Dragons class feature since I have no sorcerer levels. Do I get those bonus spells, or is it just the bloodline powers (like claws and breath weapon)?
By my reading of the ability, it only grants bonus spells to sorcerers. Non-sorcerers specifically gain bloodline powers, not bonus spells, bonus feats (except through the Dragon Disciple's own bonus feat ability), or bloodline arcana.
However, that's a strict rules-as-written reading. The intent seems to be to allow you to gain those spells at the appropriate Dragon Disciple levels, and probably the rest of the bloodline stuff as well. Not getting the bloodline arcana just wouldn't make much sense. It's probably best to ask your GM to look at it, I think he or she is pretty likely to give you the bonus skill, spells, and bloodline arcana. (You would still only gain bonus feats through the prestige class's bonus feat ability.)
Quoting from D20PFSRD
A dragon disciple adds his level to his sorcerer levels when determining the powers gained from his bloodline. If the dragon disciple does not have levels of sorcerer, he instead gains bloodline powers of the draconic bloodline, using his dragon disciple level as his sorcerer level to determine the bonuses gained. He must choose a dragon type upon gaining his first level in this class and that type must be the same as his sorcerer type. This ability does not grant bonus spells to a sorcerer unless he possesses spell slots of an appropriate level. Such bonus spells are automatically granted if the sorcerer gains spell slots of the spell's level.
Looking at the wording of the paragraph, it seems that they are using "powers" to mean all benefits of the bloodline, not specifically "Bloodline Powers". Otherwise, nobody would ever gain spells from this ability, as they are not part of "Bloodline Powers".