These kinds of questions are fun.
- The preferred path will always be counter-orbit. I don't want to move forward toward Gor because it's moving away from me and that will lengthen the mission. We want to move backward, so Gor is approaching us as we approach Gor.
In this case we have the most fuel efficient solution because all we need is to get the ship out of our atmosphere. If all it did was sit there with a relative velocity of zero, 180 days later it would collide with Gor. However, cheap as it may be, time is money! We need a faster solution, but preferrably one that doesn't cost us a mint.
Trivially, we can give our space ship constant thrust. We can leave it on the same path — the planetary orbital path — and reduce the number of days waiting by the allowed thrust period. OK, it's fast. But we can do better.
We're using contemporary technology, the cost-vs-time trade-off (which always exists) is basically the size of the boosters + the capability of the onboard engines vs. how long we're willing to wait to get the job done. Frankly, we're basically in the hands of newtonian physics anyway — but that includes the sun! If we pull the flight path closer to the sun, we actually get a boost from the sun (both in terms of acceleration and deceleration), which speeds us up without additional cost, and shortens the path.
Let's use some real data to ship some cargo. Humans will be slower because, well, we hate being squished. Therefore, let's use New Horizons as our benchmark. It left the Earth at a screaming 16.26 Km/s and passed the moon's orbit in just 8.5 hours. Smokin'! But the moon's a kicked can compared to Gor.
Now, here's where I'm very happy you didn't include the hard-science tag. I stink at orbital mechanics. I'm slowly working through a textbook on it, but I'm not to the point I can just pull the math out of my head. Not by a long shot. But, here's the gist.
We're working with a sphere (a sphere described by the orbits of Earth & Gor) and there's a reason we fly over the pole here on Earth. But remember that our launch point and destination are both on the "equator" of that sphere. therefore, the sphere actually buys us little if anything at all.
We have an initial velocity that's fixed at 16.26 Km/s. However, we'll speed up as we approach the sun. We want to get as close to the sun as our heat shields and (and this is important) the oribit of Gor will allow. That minimizes the time of transit.
We do need to miss Venus and Mercury. They're not particularly in the way, but there are certain times of the year where they might (I stress, might) be inconvenient. However, we can always lift the path of the ship on the sphere such that it flys over or under the inner planets and avoids them completely — so they're really not much of a threat.
Earth's orbital path does have an eccentricity, which I'm going to completely ignore. The biggest reason for this is that, when thinking of the orbit of Earth as the reference plane, there is no eccentricity between Earth and Gor. Huzzah! (On the other hand, the gravitational effects of Venus and Mercury might be non-zero. OK, lift the flight path higher on the sphere so we can ignore that, too.)
What would be the actual shortest path given all of the above? Well, you need the circumference of the orbit (940x106 Km) and you need the effect of the Sun's gravity (274 m/s2) and you need our reference intial velocity (V0 = 16.26 Km/s) and then [magic happens here].
What can I say. I'm not to that point in the book, yet.
If HDE 226868 or Kingledion answer, be sure to upvote them. They are initiates of the inner circle and know the correct incantations and which Demon to invoke to provide an actual, numerical answer.