The example shows the present progressive construction, of which one use is to describe an event taking place at the time of speaking, as in, for example, I’m eating my dinner now.
I am going to school now is unlikely to be used in that way. Another use of the construction is to express the future, and that is what it seems to be doing here. Your interpretation that the speaker is leaving for school fairly soon after finishing speaking is probably the correct one.
An alternative is to understand the statement as being said by a small child, proud to be old enough to go to school, not just at the time of speaking, but in general. That meaning, however, is more typically expressed by the present tense, as I go to school now.