There may be a different system for flowers (my background is more fruits and veggies), but basically the rule is that if you bred it, you can name it. (Unless you bred it as part of your job, in which case your employer would probably like to be part of the decision.) Generally you'll want to make sure it isn't already in use for something of the same (or similar) species, so you're not basically committing fraud by marketing it.

If you're going to patent it and/or trademark it, remember that you can't trademark the name it's patented under. Beyond trademarking and patenting, there may not be an official register for many species, and even if there is, there's not necessarily any requirement that you use it.

As far as tips, the biggest one I can think of for a beginner is just to be aware that simply taking a plant with trait X and a plant with trait Y and crossing them is not necessarily (or even likely) going to give you a plant with traits X and Y. Maybe you'll get lucky, but things are rarely so simple. Just enjoy the diversity that results. Certainly keep in mind your breeding goals, just be prepared for it take a few generations, and be prepared to find something totally unexpected. And be prepared to find a whole lot of worthless stuff in the process.

Here is some information on registering cultivar names, but I don't see anything specific to asters.

The Internationale Stauden-Union is the International Cultivar Registration Authority on the genus Aster. Registration of new cultivars is free.