Learn English – I wish or I do wish

Both are normal grammatical English, not slang.

A key difference is that "could have" often emphasizes a passive lack of opportunity, while "had" sometimes implies an active decision.

I wish I could have gone to your birthday party.

I could not. Why not? For unstated reasons: it was not possible, I could not have gone, I wish the universe had been otherwise, I wish I could have gone.

I wish I had gone to your birthday party.

I did not go. Why not? Perhaps due to an unstated choice: I chose not to go, I wish that I personally had made a different choice, I wish I had gone.

Note that "had gone" does not unambiguously claim choice and responsibility -- however "could have gone" clearly denies choice and responsibility.

• "I wish to be a pilot" refers to the future. You have a wish to become a pilot. Most probably you are already doing something to become a pilot; moving in that direction so as to become a pilot.
• "I wish I was a pilot" refers to the present. It's called the subjunctive mood. You use it when you're talking about something that isn't true or when you want to express a wish. The past simple indicative of "be" is correct, even the Oxford Learner’s Dictionary considers it as correct, however, "I wish I were" is the most correct form, and, if you are writing a formal piece of writing, definitely go for it.