Converting a non-planar graph to planar
These problems usually appear under the name "X edge deletion" and "X vertex deletion", where X is the graph class of interest, e.g., X could be "planar". There are also different variants where you allow for say edge addition and deletion (these are "editing problems"), or the operation could be edge contraction and so on.

In other words, you seem to be interested in both planar vertex deletion and planar edge deletion. For starters, you can see [1] and [2] (the problems seem FPT parameterized by the number of operations $k$).



[1] Marx, Dániel, and Ildikó Schlotter. "Obtaining a planar graph by vertex deletion." Algorithmica 62.3-4 (2012): 807-822. -- Non-paywalled version here.

[2] Golovach, Petr A., Pim van’t Hof, and Daniël Paulusma. "Obtaining planarity by contracting few edges." Theoretical Computer Science 476 (2013): 38-46.