Sally,

How many weeks? With 8 players it is possible for each two to have been together exactly once after seven weeks. That schedule is pasted below. Just use it over and over until your season is complete. To get a schedule for 8 players, use the 8 player schedule and delete player 8 wherever s/he occurs.

In the schedule below, the 8 players are the 8 positions in each list, and the numbers indicate the group the corresponding person is in. For example, on Day 1 the foursomes are 1, 2, 5, 6 and 3, 4, 7, 8. On any given day the groups can tee off in any order, so player 1 does not need to be in the first group out.

Day 1 : (0, 0, 1, 1, 0, 0, 1, 1)
Day 2 : (0, 1, 0, 1, 0, 1, 0, 1)
Day 3 : (0, 1, 1, 0, 1, 0, 0, 1)
Day 4 : (0, 0, 1, 1, 1, 1, 0, 0)
Day 5 : (0, 1, 0, 1, 1, 0, 1, 0)
Day 6 : (0, 1, 1, 0, 0, 1, 1, 0)
Day 7 : (0, 0, 0, 0, 1, 1, 1, 1)

Hope I understood the question.
—Victoria