While not shown in popular televised science fiction (which is all many know), there are some technologies that are real that ANY space race could use.
Question 1: How are you getting the space stuff off the planet? Gravity is a big issue you know?
Hard science solution: Make a bunch of space elevators with carbon nanotubes or similar stuff. This allows you to get stuff into space cheaply. A lot cheaper and quicker than rockets. Did I mention safer? It is that too. No more worries about getting killed on reentry. Since the elevator is connected to the ground, you won't have to reach ridiculous speeds just to escape Earth's gravity.
Question 2: How are you gonna get around the solar system without running out of fuel?
Hard science solution: Solar sails. You may want to add a little fiction, or at least add a laser assist so they won't be totally dependent on the sun though.
Conclusion: All space races should at least have this tech, plus rockets. Since it predates any of the more common types, like warp drive and so on. And I don't think a space race would necessarily dump all their old tech. At the very least, they would keep some of it if it still worked. Some for museums, and some just as a fail safe.
I don't know about you, but I want a space elevator to rely on if my fancy star trek teleporters fail.
Because I'm not in the anything goes spaceship camp for fiction. Hardscience mixed with necessary handwavium I find more attractive. Since I know the science is at least remotely possible.