[RetroTechie]

For a simple-enough design, I'd just route the signals as on a 2-layer board. Play around with placement & orientation of parts to minimize overall trace length. If you have some A-B duo with many connections between them: place those first & then put the other parts.

Then fill the remaining space with power planes. GND on one side, most-used supply voltage on the other side.

That way you reduce/avoid the need for blind (or worse: buried) vias.

Another trick: route signals roughly east-west on one side, roughly north-south on the other side, and use a via where you change direction. Many early (complex!) computer mainboards were done that way on 2-layer boards (often using through-hole IC pins or passives as via).

You shouldn't really need 3+ layers unless you get into high density stuff like many-ball BGA parts & the like.