providing i have necessary equipment could i theoretically travel in space or would i need consent from NASA o

Might need a permit for that one.

They would probably arrest you upon landing for some kind of charge like violating airspace or some such B.S.

No, you only need a permit from the FAA, they are responsible for civilian suborbital flights. When you fly into orbit, you don't need a permit by NASA, but the government will want to know what you are up to, as they will have to pay other countries for any damages you cause - and might want to know if they can expect you to pay the money back.

And the government agency managing US spaceflight is... NASA ;)

nasa don't care, but the faa do. look for form 8800-1.

NASA does not have exclusive 'rights' to space travel. If you can develop a vehicle to get free of Earth's gravitational pull, then no one can stop you.

NASA doesn't own space, nobody does. If you can come with the goods (like your own spaceship) then go for it, and good luck.

It's true that technically you don't need anyone's permission (other than the FAA for the suborbital part of your flight) but good luck getting all the materials and fuel and building the thing without having to make friends with people in high places.
Building a rocketship and building an ICBM are amazingly similar.