To achieve fuel economy during orbital launches (for a particular orbital altitude), do we select the first stage (its thrust), in such a way so as to achieve the desired altitude above the Karman line, (with all the payload - such as that of the hardware for all the stages) with the least acceleration, and then once at the desired altitude, fire the rockets (may be second stage) to impart enough tangential velocity in that orbit (e.g. 24000 kilometers /hour)? I might have elaborated more than required, but that's because my experience is people tend to comment more on the letter than the spirt of the question. - So, sorry for that.
Fuel economy is absolutely imperative in rocket launches. The rocket equation is such that we have no wide margins in getting stuff to space.
But we have some (sometimes opposing) parameters to keep in mind and balance them against each other to get the optimal launch profile for each mission.
- We want to accelerate as fast as possible to minimize gravity losses
- We want to turn horizontal as early as possible to minimize gravity losses (in the end, everything that counts is horizontal velocity for orbital flights)
- We want to leave the dense parts of the atmosphere as quickly as possible to minimize aerodynamic losses (and reduce aerodynamic stresses)
- We need to keep the speed inside the atmosphere within certain limits to not overstress the structure
- We need to keep the acceleration within certain limits the whole trip to not overstress the structure and the payload
About the karman line: It has no meaning at all for our purposes here.
So your idea of going vertical as quickly as possible to get above most of the atmosphere and than turn horizontal fulfills some points some points in the list above but contradicts with others. It would also be very hard to turn a rocket quickly and impart high loads on the structure.
So every rocket start is a question of finding the optimal parameters the get as much payload to the destination with as little fuel as possible while staying within the structural limits of the system and with the performance available from the rocket motors.