Hey Foggy,
For people in your position, I recommend
this page on A Little Weird to start out. It has links to both research as well as sites that have exercises you can try out for yourself and see your results. There are also some recommended books to buy or borrow from the library.
I personally recommend starting out with
psi balls when it comes to experiencing something for yourself. While it's certainly possible they provide a placebo effect, I don't think it's the case that they do, given the reading I've done about them and other energy work, as well as my own experiments in which friends could tell me what the one I created felt like before I told them what I thought it felt like. In any case, for me, they've always been the easiest psychic feat to accomplish, and I've built my base around energy work since then.
Edit: Whoops, missed the second part of your post. Spirits and psychic abilities can exist in an objective reality as easily as they can in a subjective reality. The explanation for why they exist could change, but they'd still exist. Also, just because science hasn't found something doesn't mean it doesn't exist. Einstein came up with the theory of relativity a little less than a hundred years ago. That doesn't mean relativity didn't happen before, just that no one understood certain things about the world yet. The same could just as easily be true of psychic abilities, except we still don't understand them very well. But someday, we likely will.
As for my own experience, I started out skeptical, and remain skeptical. By skeptical, I mean I refuse to believe something just because someone says it's true. With a convincing explanation, I'll give it some thought, and with evidence, I'll likely believe. Skepticism provides a defense against charlatans, which do exist amongst the real deal.
My interest in psychic abilities has always been there. I've always been fascinated by magic and fantasy. I just didn't think it was real. Then I saw a talk show with some psychics on about ten years ago, a mother and daughter, though I don't remember their names, and they claimed that anyone could learn to be psychic. So, I started looking for things online that taught how to be psychic, tried astral projection with no or very limited success, I'm not sure which, and tried a lot of remote viewing types of exercises with no success.
Then I came across energy manipulation. I had immediate success creating a psi ball, and when I found some like minded friends, they could confirm that they felt the same things I did. From there, I've been practicing lots of energy manipulation, though I've found myself still quite stumped on how to learn perceptive abilities like remote viewing and telepathy. No luck at all with telekinesis, and I've dropped my focus away from that for now, since it seems the most outlandish ability and is therefore the hardest for me to justify putting effort into when there is so much else to learn. As for astral projection, I think I'm on the cusp of doing it successfully, but time will tell on that.