We know that Gor, or Counter-Earth, is not stable in a long-term sense. Let's handwave that away, or assume that we are dealing with a temporary span of time in which the instability is immaterial.
Mid-20th to early 21st century Gor and Earth have finally discovered each other and made contact. An alliance of nations on Gor have succeeded in building a space station in Low Gor Orbit (LGO) and are very interested in starting regular passenger/cargo shuttle flights to Earth's International Space Station.
What trajectory would a more-or-less contemporary crewed spaceship use to travel between Earth orbit and Counter-Earth orbit? We are dealing with civilizations that are, technologically, more or less anywhere between Earth's mid 20th century and a few decades from now. That is, incremental advances in technology (e.g. slightly more accurate thrusters, 10% more effective radiation shielding, or tastier meals-in-a-pill) are fine, but no Stargates or Millenium Falcons. The trip should be as short as reasonably possible, but should keep the persons on board reasonably safe. Spaceman Thumper reminds you that if you fly directly into the Sun, you are going to have a bad time.
I'm imagining that ships would simply drop into an elliptical orbit that would put them onto a trajectory to intercept the other Earth, but what would that look like? Would such an orbit be quick and reasonably achievable, or are our hypothetical cross-world pioneers going to be wandering around the Solar System like Odysseus?
Economic issues (exactly who pays for what, who gets to purchase the first interplanetary tickets, etc.) are out of scope for this question unless there is a particular reason they need to be included. Assume that governments on both planets are treating this as a high priority project and are able to provide at least as much funding as modern-day space programs receive.
As for spacecraft design, I am imagining something reasonably similar to modern designs, with the obvious caveat that such an interplanetary shuttle won't need to take off or land and so could forgo much of the equipment found on, e.g., the Space Shuttle or Soyuz. Instead, such a ship could that space for whatever it needs to maneuver in space or keep the occupants alive. Modern designs for non-atmospheric craft that have never been built but appear spaceworthy are obviously acceptable.