This is a good question. For people from my generation, these 3 phrases mean the same thing. For young people, they might have a variation in meaning. Good luck!

I think the difference reflects the uncertainty of a new relationship. If you say you're dating someone, then it's official. They are you're boyfriend/girlfriend. If you say you're seeing someone, it could be something new and you're unsure where it's going, or maybe you just don't want to admit it. This phrase would often be used in the third person. "She's seeing someone, but we really don't know how serious it is." Going out with someone is mostly used to mean dating. It's not very ambiguous, but it isn't as strong as "dating." When I was a kid it was in school it was a euphemism for dating.

Yes, they are all similar in meaning.