A spacecraft is travelling between planets. It accelerates at a constant rate for the first half of the trip, and then decelerates on the other half. Throughout the journey, aside from the switch in direction in the middle, the occupants of the ship experience a comfortable consistent gravity. Longer trips are maybe pieced together from several pairs of acceleration/deceleration legs, such that at no point does the spacecraft have to deal with particles hurtling at them at relativistic speeds.
Instead of dropping the entire ship down a gravity well and hauling it back out, smaller ships (better equipped for reentry and other aerodynamic shenanigans) are used to transport passengers and cargo between spaceship and planet. The ship waiting in orbit moves a little faster than a free object in the same orbit otherwise would, and provides the necessary extra centripetal force by continuously firing its thrusters in the direction away from the planet below. Crew waiting aboard the ship experience artificial gravity with “down” pointing towards space. (Or vice versa, with a slow ship and propulsion away from the planet instead of into it, and artificial gravity in the opposite direction.)
A space station with artificial gravity produced in the aforementioned fashion. A space elevator connects it with another station in a higher orbit, which experiences weightlessness because it is orbiting with the same period as the “bottom” station. The “top” station provides a controlled weightless environment, perhaps for all kinds of zero-gee science, while a permanent/long-term crew are rotated through the “bottom” station so that they experience gravity often enough to keep their health from crashing. Traffic between the station and the planet below carries cargo (food, propellant, other necessities) exclusively.
The idea is to remove the problem of adapting to weightlessness, simply by having every tin can in space that has people in it move in a way such that gravity is experienced nearly all of the time. This question is about the tradeoff between the cost of propulsion (propellant, and the maintenance of engines that are kept on 24/7) and the convenience, and other benefits, of having gravity.
Because burning expensive chemical fuel just so things don’t float away some of the time is extravagant to an appalling degree (and simply isn't sustainable anyway), we might suppose fusion (or some other method of getting a lot of cheap energy from very little mass) is available.
edit: the intention of the previous paragraph, and the wording of the following question, had been to indicate that the problem of generating the sustained high thrust required for the implementation of these schemes is to be taken as issue that can be circumvented. They were written based on the assumption that access to cheap energy implies the capability to generate those thrusts; as the answers from Rekesoft and Firedrake below have pointed out, that assumption is not true.
Nevertheless, I am still interested in being informed of other problems that are relevant to the scenarios described, now supposing that some sort of magic space drive is employed.
What other obstacles must be overcome to have widespread use of artificial gravity be practically favourable? What other important things am I overlooking?
Some more things I’ve considered:
I don’t think I’ve done sufficient research on ways to alleviate the detrimental effects of microgravity on human bodies to judge this, but I suspect just having some gravity (say several tenths of a gee) might already be very helpful. In that case, doing some simple maths and taking data from some wiki pages, It seems like having around 0.2 gees would be, considering the thrust required kilogram by kilogram, about four times as expensive as keeping an airliner in cruising flight. I have no idea how economically sound this would be, but feel like it may not seem too unreasonable if propulsion is sufficiently cheap.
Possibly relevant is this question, in which artificial gravity is achieved through the same mechanism in a ring habitat, except the load is supported by structural integrity instead of propulsion.