Which Richard Osman book should I start with?

There is no crossover you need to understand: the two series are independent, and both welcome new readers.

The short answer

Begin with The Thursday Murder Club for a close-knit ensemble and a gently paced village setting. Begin with We Solve Murders if you want a smaller central team, more travel and a brisker thriller rhythm.

Should I read out of order?

Each mystery resolves its central case, but relationships and personal histories develop from book to book. Publication order gives those quieter arcs room to land.