Buying a property and moving kids to Bulgaria

My suggestion is to search the archive on relocating to BG. There is a lot of information there. Ease of relocating depends on your current location and what your game plan is once you are in BG. In the archives you will also find a lot of information on buying property. Prices are higher near the sea.
The general recommendation is visit multiple times in different seasons to determine if you actually like BG. As for schools, the same recommendation applies as above. There was an English language school in Burgas, but I am not certain that it it still operating.

Assuming you are British, residence permits are relatively easy during this year (the "transition period"). I'd recommend sorting out residence permits during 2020, rather than leaving it until later.

All big Bulgarian cities (Sofia, Plovdiv, Burgas, Varna) have private schools. (But "big" by Bulgarian standards is quite small by UK measures.) I'd suggest that if education is a top priority (which it probably is) that you give serious consideration to Sofia (and maybe Plovdiv) even though it's not on the coast. You could, for example, consider a bigger home and a better school in Sofia, and a holiday apartment at a Black Sea resort.