Orbital mechanics question (around sun)

In a Mars expedition scenario, would travel between earth-mars and back
be timed so that it can be done almost radially (relative to orbit
around sun) ?

If a ship leaves mars and heads straight for the sun (with Earth planned
to be on its path), will orbital mechanics relative to the sun come into
play ?

Or would such a scenario involve first leaving Mars's gravity, and then
slowing down relative to solar orbit to drop back to the orbit used by
earth ?

Or would they time such a trip so proecisely that the difference in
orbital speed resulting from radial travel between solar orbits would be
nullified by the gravity of the receiving planet ?


Or is the solar orbit/gravity fo sun inconsequential for such travel and
only affecting a few decimals at the end of some orbital formula ?



If solar orbit is of consequence, would it be possible to park a
resupply ship in solar orbit above earth's orbit timed such that the
returning expedition ship could dock with the resupply ship ? The
resupply ship would only need to accelerate to match the returning
expedition ship's speed vector.