seating arrangement for 4 handed game

do partners set directly across from each other in 4 handed cribbage so that my partner plays next after me

Partners seating

Usually partners sit diagonal from each other, so that one can peg and the other does not. Your partner does not start out playing right after you, but on the count to 31, they might if the other team has exhausted their cards.

Doubles Cribbage

Also, of note, in doubles cribbage, your partner is normally also allowed to view and correct your hand score before it is scored so that you can agree on the points before you move the pegs. This really helps out so that points are not missed.