"I'm doing something" (remember to add a future time!) means you have an arrangement. Basically, you and others (I mean, those involved) know. "I'm going to do something" is your own plan. You probably haven't made any arrangements with other people.

"I'm doing something" is the present continuous. The present continuous expresses an activity that is in progress. "I’m going to do something" is the simple future tense. The simple future tense expresses an action or situation that will occur in the future. "Going to" is used to express the simple future tense in English.