[mabbo]

> For example can we use a space elevator to start higher/faster (or, I don't know, balloons? a catapult or railgun or something?), laser power delivery from the ground, so we don't have to carry all the fuel, and an orbiting way-station for refueling, etc.?

These methods will all help with the first 1% of your problem, getting off of the earth.

But you need so many orders of magnitude more energy to reach the kinds of velocity needed to get to another star in less than a million years. It's just an unfathomable amount of energy per kg. Put simply: if you can get to another star, getting off the planet is nothing.