about a spacecraft trip to the asteroids

This was something I was kicking around and wanted some advice on.

Let's assume you had a big spaceship -- the kind you'd use t set off on a trip to the Asteroids, and were willing to take some time about it, on the order of a couple of years. Maybe you want to do asteroid mining or mine iceteroids for volatiles.

You're in Earth orbit, where the thing was built. You have a nice, big, NERVA type engine.

Which is the better plan, with the best combination of shorter travel time with energy efficiency:

-- Increase the ship's orbital speed around earth -- let's say it starts in MEO - and get it to escape velocity in the direction retrograde to Earth's orbit. That will slow it down relative to the Sun, and the ship can fall in and slingshot around to the asteroids. When the ship is deep in the sun's gravity well you could even do a burn and boost its speed further, and get a nice long parabolic / extremely elliptical trajectory to the relevant spot.

-- Same thing, but going prograde, and using the free velocity you get from being around Earth and traveling with it to boost out to Mars and past it, since the delta-V would be (I think) less to get the same nice orbit out to between mars and Jupiter, which is approximately where you want to be.

-- Some kind of VEGA trajectory.

I was also curious about how to set of a good langrangian for this. Was thinking of using the sun as my origin, but if anyone has a better idea please!

(And I was thinking of modeling this on Mathematica for giggles, but I don't know if it would work at all well as I am a beginner)