here is a website dedicated to the ideas and philosophies around Dragon magic and working with dragons in general.
Bear in mind that Dragons work with you and within your own practices. It's one of the reasons why people tend to have different experiences with them, and come to differing understandings of how dragons work with the people they choose to take under their wing. (yeah... puns... ¬_¬)
Here are a few basic principles;
The term 'dragon magic' needs a touch of perspective. It tends to bring to mind that there is some unique energy or method or practice specific to dragons, but this isn't -quite- right. it would be more accurate to think of it as you using whatever magical practice you already call familiar, and engaging the assistance of dragons within your own workings.
For example, if you call corners, instead of calling on deities or elemental archetypes, call for a dragon to come as an elemental master and watch over/govern the corner. Dragons, as ascended beings, all have the same capabilities of working in energy and, just like with other beings and guides, develop individually into different things they specialize in.
Being habitually ancient beings and highly experienced, they tend to connect with more primal (simple, pure, potent) forms of the energies. So if you are not used to that sort of intensity of energy flow it can feel a bit overwhelming or jarring at first. But it also means that a lot of change tends to get stirred up both internally and externally when they are around.
As ascended masters, they tend to act as guides and guardians. They rarely act to lend or grant their energy/power to the magic worker. Instead they help the worker find and connect with their own power. As you work with them expect a lot of introspection, self reflection, and all kinds of little nudges and revelations. This isn't to say that Dragons never act as a source of energy or direct influence, or otherwise work directly through yourself as a caster/energy worker. But it isn't done casually. I've found that most often, even while a Dragon is allowing you to connect with and use their energy they are doing it to teach you something, or to help you form your own connections and familiarity with the sort of vibration and quality of energy they work with naturally.
When it comes to the sorts of personalities and interactions you can expect, it is difficult to really offer any specifics. dragons are very individualistic critters, much like people each is his or her own in attitude. But, just like when meeting other people the dragons that come to you tend to do so because they are of an attitude that is more or less compatible with what will most efficiently help you to learn and grow. Some people need to be challenged a bit, so the dragons they connect with might be a bit more aggressive and demanding. Other people might be studious in nature but have a tendency to get stuck in a rut or series of habits, so the dragons they work with might be more likely to show up and give them a swift kick in the butt to knock them into gear. Some dragons might be more impish or playful, or subtle and poignant, or patient and quiet. That sort of thing.