If he's not your father, a Father's Day gift is not required.
If you're looking on behalf of his child (your child together or his children from a prior relationship), make it something age-appropriate.
Birthday gifts: Home cooking is always a great idea - I usually splurge on the best steak I can afford and bbq them (served with his favorite sides and a nice homemade dessert afterward). Our local gas station sells car wash cards that are good for 90 days (1 per day) and Mr Scully loves that.