So my first name is hard to pronounce, EVERYBODY screws it up. A couple of my close friends call me "Vee", but the rest of the world knows me as Vadim (Vuh-deem). I'm wondering which name I should give out...

Vee sounds like an awesome name (although I would write it as V). Have you seen V for Vendetta? Watch it and you will like the name.

So yea, go with V.

Which one are you more comfortable with? Which one makes you feel closer to people and them feel closer to you?

I generally introduce myself as Rye, unless it is for something where my real name is going to be used often, like school or work. What I will do though, is tell people my real name and then say, "But you can call me Rye; all my good friends do." This instantly plants the idea that we are good friends, or going to be good friends. Makes them feel closer to you and especially because your name is hard to pronounce (doesn't seem like it to me, but you sound brown and I've known a lot of brown people over the past 5 years, so I'm probably just used to it), they'll probably go with the nickname.

_________________
"The 'Brick Walls' are there to allow you to prove how badly you want something!" ~ Randy Pausch

~ Rye