This sounds like a classic usecase for the EXISTS
operator:
SELECT *
FROM mytable a
WHERE EXISTS (SELECT 1
FROM mytable b
WHERE a.foreignkey = b.foreignkey
AND trainer IS NOT NULL
AND trainer <> 'FREE'
EDIT:
If you just just want the distinct different foreignkey
s:
SELECT DISTINCT foreignkey
FROM mytable a
WHERE EXISTS (SELECT 1
FROM mytable b
WHERE a.foreignkey = b.foreignkey
AND trainer IS NOT NULL
AND trainer <> 'FREE'