What would rockets look like if the Earth had no atmosphere?
How much easier would it be to launch satellites, and how much harder will it be to launch returnable manned missions?
Atmospheric drag and gravity losses account for about 1.5-2km/s of the normal 9.4-10km/s launch-to-LEO budget.
With the trajectories we use, the atmospheric drag accounts for only a small portion of that, but we'd use more efficient trajectories and lower orbital altitudes if we didn't have to worry about the atmosphere! So at a guess, we'd need 8.5km/s instead of 10km/s of ∆v, which because of the nonlinear nature of the rocket equation, suggests you'd need only about 55-60% as much rocket for a given payload.
However, the design of Earth-launched rockets is heavily constrained by the atmosphere. They're made tall and skinny to minimize drag, and have a streamlined skin around their tankage and payload, all of which adds weight. A rocket made to launch from a vacuum world would be made mostly of big spherical tanks attached to a minimal frame. The more compact shape would lead to a lot of miscellaneous weight savings -- shorter cable runs, propellant lines, etc -- which, while individually small, would add up.
Launchers could use efficient, light-weight, low-pressure, liquid hydrogen fueled engines from the first stage, as well; system specific impulse gets better, rockets get lighter, everyone's happy.
Return missions would be just brutal, though. In our manned missions, atmospheric braking provides essentially all the deceleration from orbital speed. Without the atmosphere, if you needed 8.5km/s to get up, you need 8.5km/s to get back down in one piece. To land a one man capsule like Mercury you might need to get something like 50 tons of payload into orbit!
If Earth had no atmosphere I think you would see a far more radical change to spaceflight that the rocket redesigns other posters are talking about. While rockets might have been used at the dawn of the space age I think we would be using something akin to maglev trains for launch. Take a big flat area and put a very, very long track on it. Unlike a train it would boost at rocket-type accelerations. You boost your craft until it reaches the velocity of an orbit with perigee at 0 and apogee at the desired altitude. The craft heads off into space, the train slows and is recovered. When it reaches apogee it does a circularization burn.
If your accuracy is good enough you can even use a similar system to land--although I would do so with cables to grapple the returning spacecraft rather than attempting to actually dock with the train.
Fundamentally, the atmosphere of the earth affects the launch vehicle in two ways:
A simple rocket equation shows how the thrust produced by the rocket is used to overcome the forces acting against it. The thrust, $T$ can be given as,
$T = ma + mg + F_D$
where m is the mass of the rocket(and payload), a i the acceleration, g is the acceleration due to gravity and $F_D$ is the drag force acting on it.
If there is no atmosphere, the Drag term will not be there and the excess thrust can be used to increase the acceleration. Also, because there is no atmosphere, there is no need for a heat shield and this further reduces mass of the rocket. So, basically, we can reach higher altitudes with lesser fuel or use less fuel for reaching the same altitude.
As for landing, it is instructive to see how NASA landed in MARS, which has about 1$\%$ of earth's atmosphere.
For landing the Phoenix in MARS, NASA had to use a two stage system- a parachute first slowed the spacecraft down and then retro rockets were used to land it.
If there is no atmosphere, it is useful in one way because there is no reentry heating of the spacecraft; however, the only way to land will be using retro rocket thrusters.
Photo Credit: NASA/JPL/Corby Waste
This will make the system much more complicated. Note that I'm comparing only the usual single-use systems, not reusable space systems like the shuttle.
So, if we were to consider only the launch/recovery, if there is no atmosphere, it will be comparatively much easier to launch the rockets into space while it will be as much if not more difficult in the return leg.