For two players who like to plan ahead, start with one of the traditional strategy games.
Nine Men's Morris goes back at least to the Roman Empire.
Fanorona comes from Madagascar.
Tigers & Goats, the Bagh-Chal game from Nepal, is lopsided on purpose: one player's four tigers hunt the other player's twenty goats, and the goats win by trapping the tigers.
Five Field Kono is a Korean game where you race to move all your pieces into your opponent's starting spots. Two more: the
Fifty Game has you leapfrog pegs toward the 50 hole, and in
Linex you turn octagonal pieces an eighth of a turn to complete a line of three. We make Nine Men's Morris, Fanorona and Tigers & Goats in our Hudson, Florida workshop, each with a wooden cover so the pieces stay with the board.