Talking to Aeris:

        Go together                 (+3 Aeris)

        Don't go together           (-2 Aeris)

    Talking to Tifa:

        Go together                 (+3 Tifa)

        Don't go together           (-2 Tifa)

    Talking to Yuffie:

        Go together                 (+3 Yuffie)

        Don't go together           (-2 Yuffie)

    Talking to Red XIII:

        Go together                 - No change -

        Don't go together           - No change -

Only the first of the above Affection changes may be made: if you decline Aeris first, you'll get -2 Aeris, but you will be unable to get a +3 Tifa as well, for example.

You can decline Red XIII for no effect and still get an Affection change by talking to one of the others.  If you ask Red XIII to come along though, the rest of the Affection changes are locked off.

If you leave the area without choosing a partner, the character with the highest Affection will automatically join you, and no changes will be made.

If Barret has the highest Affection, then Red XIII will join instead.