Welcome to the Forum and Orthodox Christianity.
When I first contacted a priest I simply gave a short summary of how I was introduced to Orthodoxy (it was through books for me) and then said I'd like to set up a meeting to talk. If you don't feel comfortable with saying you want a meeting, just make the initial contact via email and say you've discovered Orthodoxy and are interested in it. Any decent priest will of course offer a meeting at this point.
I too was afraid of meeting a priest, but someone told flat out "don't worry about kissing his hand and titles just shake his hand as you would any normal person". That's excellent advice. Don't worry about the ettiquette, it's irrelevant for you at this point. He may be a priest, but in the end he is just a human being, just a guy, nothing more. He has no magical powers, no special connection to God that you yourself cannot also have; just a guy. Again I know it might feel really scary and overwhelming at first, and I'm sure if when I was in your position someone told me what I just told you, it probably would not have mattered much, I'd still be scared, but I've been Orthodox long enough to know priests and Bishops are just people. Some good, some...well, just like everyone else. Don't concern yourself with all that at this point. Just give a summary of how you came to know of Orthodoxy and click "send"....you can even say you are a bit nervous if you like, he will understand. If he doesn't, then you'll know to try another parish with another priest. Though I doubt that will be the case.
Just relax and be honest, and I hope your meeting goes well.
Oh yes, you CAN "just show up" to Liturgy on some Sunday morning as well. No need to ask permission to come to Church, it is after a Church.