There are two major design types of liquid fueled rocket engines: pump-fed and pressure-fed. These terms refer to the way that fuel and oxidizer are forced into the combustion chamber. Pump fed designs, such as the SSME, use high-pressure turbopumps to generate about 3000 PSI. High chamber pressure allows these engines to have high Isp, and the total system is relatively lightweight and highly efficient. Unfortunately, the pumps are expensive and complicated, require a lot of maintainence, and can only be considered lightweight when compared to the alternatives.
The other design is to have the fuel and oxidizer in a big, sturdy pressure tank and keep the tank at high enough pressure that it naturally pushes into the combustion chamber. This is mind-bogglingly simple and requires virtually no maintainance. Unfortunately, large pressure tanks are very heavy and the chamber pressures which you can maintain are much lower, which leads to lower fuel efficiency (in terms of Isp). Most Russian rockets are very robust pressure fed designs. Most American rockets are more efficient pump fed designs, but suffer from extremely high maintainance and production costs.
Getting to orbit is mostly a problem of going fast enough. In these terms, the energy spent getting to altitude is just losses in the energy spent going fast - "gravity losses". Likewise, the drag induced pushing though the atmosphere are drag losses. Also, rocket nozzles optimized for vacuum don't work as well in atmosphere, because the air pressure pushing back into the nozzle.
Take a helicopter. Pull the engine out. Run fuel lines down inside the length of the blades. Put rockets on the rotor tips. Use the centrifugal force on the fuel in the spinning blades to pressurize the combustion chamber to very high pressure. Use the rockets to turn the rotor. Throttle the rockets way down to conserve fuel, and use the aerodynamic lift of the rotor to go up. As the air gets thinner, increase the collective pitch on the blades for more lift. This also increases the downward component of thrust from the rockets. At the same time, gradually increase the propellant flow to the rocket motors. By the time there is virtually no atmosphere, the rockets will be pointing almost straight down (although not completely, since the blades have to keep spinning in order to keep the pressure up). The downward component of the rockets will continue to propell the craft.
Compared to a conventional rocket, drag losses are vastly reduced since you don't really accellerate until you're already out of the atmosphere. Gravity losses are handled aerodynamically, which is much more efficient than rocket engines. Sea-level Isp for the rocket engines still sucks, but it doesn't matter, because they are throttled way back, and the effective Isp of the entire rotor system in atmosphere is very high. It's basically a pressure fed design, but the pressures only have to be high enough to get the fuel from the tank up to the rotor head. The rotors act like a large, built-in turbopump.
Re-entry is tricky. If the disc loading is low enough you can probably autorotate high in the atmosphere and the heat won't be too bad. You can use the fuel channels to do active cooling of the blades.
There are some engineering challenges, such as the design of a rotor head with an integrated bipropellant rotating fluid coupling that is LOx compatible. While difficult, this is, by no means, an impossible problem. The one truly intractible problem may be the material requirements for the rotor blades.
The rotor blades must have the following properties:
Hopefully, modern composites will be adequate, but detailed analysis may prove otherwise. More information is needed about the mass of the motors, the aerothermodynamic loads, and methods for oxygen-proofing composites.
Furthermore, more information is needed about the effects of continuous high lateral accellerations on rocket combustion.