What does a sport pilot have to do to get a private pilot certificate in the US?
If you hold a Sport Pilot certificate and are looking to upgrade it to a Private Pilot certificate, your best bet is to use the requirements of Part 61 Subpart E. All of the flight time you logged earning a Sport Pilot certificate applies towards Private, but additional hours and types of flight time need to be logged.

The requirements for each are paraphrased below. You'd have to log the difference, which is a minimum of 20 hours, including another 5 hours of dual and 5 hours of solo. There's a lot more to it than what I reference below, but this is the basics.

How much of the training will have to be re-done? This should vary wildly from pilot to pilot. You'll need to redo training as required to meet standards. If you're transitioning right away, theoretically none, but it's likely that some retraining will be required.

Does it depend on the type of your flight instructor's certificate? Yes.  If the instructor has only a sport pilot instructor certificate, none of the dual instruction hours are able to be counted for the private pilot dual hours criteria.  They are not an "authorized instructor" for private pilot privileges.  If the CFI had a CFI-airplane certificate then those dual hours working towards the sport pilot certificate will count toward the dual hour requirements of the private pilot.

See the Kern INK Interpretation

Required for Sport (citation)

20 hours of flight time, including at least: 
15 hours of flight training from an authorized instructor and at least:
5 hours of solo flight training in the areas of operation listed in §61.311


Required for Private (citation)

40 hours of flight time that includes at least:
20 hours of flight training from an authorized instructor and: 
10 hours of solo flight training in the areas of operation listed in §61.107(b)(1)


If you've chosen a flight school that operates a Part 141 program, you get to do it all over. Hooray!