I believe the following sentence is grammatical and means that the person will move ahead with the planning (i.e., keep working to fulfill the plan) next week:
I'm busy this week, but I'd like to move on with the plans next week.
I don't know, though, if it'd mean instead that I'll do work on the plans (instead of in fulfilling what the plans set out to do) next week.
But does this sentence make sense? (even if only informally)
I'm busy this week. I'll move on this [the marketing campaign] next week.
Or should it rather be "I'll make a move on this" to mean that I'll go back to this activity next week, presumably starting it again after a long pause?
In general, I think my main question is: can you use "move on" + something? What are acceptable cases and what does it mean exactly?