Both are used. "Plan to" is followed by an infinitive, "plan on" by a gerund, e.g.
1. I'm planning to move to Manchester next year.
2. I'm planning on moving to Manchester next year.
I'm not sure there's a noticeable distinction between them, for most speakers. Perhaps the gerund puts a little more emphasis on the action itself rather than the object of the action; but that may well be simply my personal interpretation.
(Maybe other members will have a view!)
Student or Learner