– could the space shuttle just fly into space like a plane, rather than being propelled vertically by rockets?
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Early designs for the SS actually did intend for it to be Single Stage to Orbit, with a novel dynamic nozzle system. Essentially a rocket nozzle is most efficient at a particular external pressure. Since there is a dramatic difference between the pressure on the ground and the pressure in space, either the system uses multiple rocket stages, or the nozzle diameter is adjusted. However, this proved to be too complicated and it was scrapped in favor of 2 stages.
The other benefit of 2 stages is that you can reduce the weight prior to starting the second stage. A single stage system has to carry all the fuel necessary to get to space, and that’s extra weight in tanks once it gets there.
Rockets generally go straight up initially and then turn horizontal to enter and orbit. You could take off like a plane, but there would be a lot of wasted energy traveling a longer distance through the atmosphere. It would only make sense if you had a plane with an air breathing engine, ie a jet, and then activated the rocket at altitude. With rockets you want to get out of the atmosphere as soon as possible to reduce drag, so you go straight up initially.
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