What actions does the wizard get automatically?
A Wizard gets Channel Power, and all Petty Magic spells for free. As well, the Wizard gets some advanced skills for free, like Channeling. The rulebook mentions these, I believe under the character creation section.
The wizard chose only spells for her actions, is that OK?
Yes, that is fine.
When buying skills, they are just training them not buying them, and only train ones the have in their career set. Is this right?
It depends on the skill. If it is a Basic skill, or an Advanced skill they have already/automatically acquired, it is training. If it is an Advanced skill that they have not acquired yet, then it would cost to acquire it first, and then another to train it.
Can the wizard train channeling?
Specializing does what exactly?
Specializations provide an additional fortune die when using a skill in a manner which is related to the specialization. For example, Franz is trained in Observation and has a specialization in Sight. If Franz is asked to make an Observation Check, and it is sight-based, Franz will gain an extra fortune die to his dice pool. If the Observation Check is based on hearing/sound, then Franz would not receive the extra fortune die to his Observation dice pool.