When I create a PowerPoint with the 2010 version at home on the laptop, how do I have to save it to open on the school PC in the 2013 version?
Can I just save it or take note of a specific format or embed it with fonts?
Your presentation in version 2010 runs without any problems under all newer / higher versions. The newer build yes in their scope on the "old" 2010er version. There are no difficulties there.
The other way round, you work with a higher, newer version - e.g. 2013 -, then you have to keep in mind that this contains some innovations of various kinds, which logically, your "old" can't have. To get around this, you have to remember to save to a lower version in the newer version. You can also open this with the appropriate version and continue editing.
I hope so with the success.