Recently I have become obsessed with the idea of twin sapient species developing on separate planets of the same horseshoe orbit. As I was considering the technological advancement of this planet, I realized that as one planet discovered space faring technology they would most likely attempt to travel to their partner planet. But considering the way that horseshoe orbits work, it seems less than likely. Horseshoe orbit mean that 1) the planets are only close once a year and even then for a very short time and 2) when they are close the speed change of the planets would make it difficult to chase after. Would a culture with 2010 level technology be able to successfully travel to and land on their partner planet? If not, why would they fail?
Well, we did travel to the Moon with 1960s (and some even 1950s) technology. We also had reasonably detailed plans to use the same underlying technology to make a year-long Venus fly-by mission in 1973-1974, but as we know, we never made the trip. So technology on its own likely wouldn't have been the limiting factor in a mission such as the one you propose.
A Saturn V booster packed quite a lot of $\Delta v$ (delta-v, velocity change) capability, and with sufficient motivation, it certainly wouldn't have been beyond us to build and launch several of them for a single mission. (That was one of the original ideas for how to get the US to the Moon: Earth Orbit Rendezvous. They went with Lunar Orbit Rendezvous in the end.) The tyranny of the rocket equation would have been problematic, but we probably could get to at least 15 km/s or so with just chemical rockets if we really wanted to. The return trip might actually be easier than the outbound trip. So delta-v budget likely wouldn't have been the limiting factor. Mid-course corrections can account for smaller errors, and are a common thing to do in spaceflight. (The earlier you make a course correction, the less delta-v budget is expended in the correction because the change has more time to have an effect.)
Generally speaking, when going places in space, it isn't so much a problem that the target is moving quickly if you can just calculate where the target will be when you get there. That's no different from going to the Moon, or to Mars, or for that matter to Pluto. Computers are good at solving that kind of problems, so our ability to end up in the correct place at the correct time likely wouldn't have been the limiting factor.
Since we have now concluded that none of the major obstacles -- technology, delta-v or navigation -- are likely to be factors that would limit our ability to reach such a planet, it stands to reason that we would be perfectly able to mount such a mission, given an incentive to go there. Depending on the specifics of the orbits involved, it might take some time to make the trip, so we'd likely start off by sending an autonomous probe rather than a manned spacecraft, but there wouldn't be any magic involved.
It would be fairly straightforward for a species with a 2010 level of technology to travel to the partner planet.
There are a couple of misconceptions: The planets are not close only once a year (but when they are close, they remain close for some time). The speeds of the planets are very similar, so little chasing is required.
If two planets are in horseshoe orbits with respect to each other, then they are in nearly the same orbit, travelling at nearly the same speed. One planet is slightly closer to the sun and so moves faster. Over perhaps several hundred years it moves close enough for the planets to gravitationally interact, the planet on the inner orbit is speeded up (and so moves out) the planet on the outer orbit slows down and (so moves in).
It is a surprising feature of orbit that speeding up a planet causes it to to move away from the sun, and so slow down. The two planets swap orbits, and now the planet that was on the faster inside orbit is moving more slowly and lags further and further behind its partner. It is many hundreds of years before the planets are close again.
It would be easiest to transfer when the planets are close, as they have a very similar speed, the transfer can be done with a small delta-v. Even when the planets are distant from each other, the transfer is no harder than an Earth-Mars transfer. If you know that there is a thick atmosphere that makes slowing down on arrival easier, and perhaps friendly natives to greet you it gives lots of incentive to make the trip.
Yes. As always with space travel you cheat, by finding an alternative path that is easier than the intuitive one.
The direct way is to launch while the planets are closest, then spend an awful lot of fuel to change your velocity to match the twin planet that's moving in the opposite direction. This is difficult to do. And it's unnecessary.
In space, if you need to get from one place and velocity to the same place and velocity at a different time, you swing around another body to "waste" some time. You use another body to leave your planet's orbit, and from there you use relatively small velocity changes you swing around the solar system until you are in the right place at the right time. So you trade time for energy, because energy is expensive.
If you needed to enter a stable orbit at a different distance from the sun, e.g. travel from earth to mars, you'd need much more energy.