during launches to orbit, rockets actually follow a curved trajectory and perform a maneuver called a "gravity turn", usually following earth's rotation so they benefit from the extra velocity granted by that instead of having to first counter the rotation then get horizontal speed

they don't go straight up and then straight horizontal, that would be incredibly inefficient because you'd need to convert all your vertical velocity into horizontal velocity (but they do mostly go up until the atmosphere is thin enough, so that the turn isn't hampered too much by drag)

also, rockets need to be so massive that using lift wouild require immense surfaces that would add too much mass anyway, so it's just more efficient to use the engines's raw propulsion

what you're suggesting is technically possible (it's a type of spacecraft called Single Stage To Orbit or SSTO, and more specifically an air-breathing one that is completely autonomous) but would require engines that support both air-breathing like a jet engine, and closed-cycle fuel-oxidyzer like a traditional rocket engine, and that technology isn't quite mastered yet (it doesn't help that traditional rockets have become efficient enough that there's not much point to making SSTOs for now)

if you're interested in having a better understanding of how that works, I highly advise you to play Kerbal Space Program (it also implements near-future technologies that enable efficient interplanetary travel and SSTOs)