Depends on speed power etc...
If your ship makes a slow controlled final descent of 30 minutes to an hour from outer atmosphere (to avoid heating up in the atmosphere) and assuming that the atmosphere does not alter the way it works, your passengers would gradually feel heavier or lighter (depending on what gravity inducing devices you have on this ship and the relative gravity of the planet) until their weight feels just above or below what their actual planetary weight would be (depending on the relative gravity of the planet) at which point I would assume that the gravity devices would be turned off resulting in a small but noticeable final shift in gravity.
In this scenario (again assuming a slow ascent to space to avoid atmospheric friction) the effects would be the same but in reverse.
If your ship uses the current method of using atmospheric friction to slow down then it would depend on the other advancements in technology. You could say that thermal insulation, inertial dampening and shock absorptive materials have come so far as to reduce a planetary descent of a few minutes to some external noise and a bit of vibration, something akin to turbulence on a plane or you could say that the descent is actually pretty jarring, requiring everyone to be strapped in to various types of harness. Gravity here would equalize in an accelerated version of the above but people might not even notice it if the landing is so jarring.
In space it would be dependent on how good the engine actually is. If the engine were made so well that everything works perfectly there should be no noise and no vibration as nothing would be out of kilter, if it is more rough and ready the engine would make noises which would echo throughout a ship unless dampened by something and assuming a metallic hull the vibrations caused by the engine would be present all throughout the ship, again, unless dampened by something. Strain on the engine would take a similar course, if made well there would be no difference between flying at full speed in space and flying at half speed, if not then there would be vibration and noise.
The kind of engine you are thinking of sounds like a washer-dryer to me, a washing machine with a separate tumble dryer is better because each machine is made for its intended purpose but combining them puts unusual stresses on different components causing the life cycle of the machine to diminish.
I would suggest that while this engine is capable of both atmospheric and space flight that the preferred environment is space and that most advanced civilizations have something akin to space elevators. The atmospheric capabilities would be used only in times where there is no alternate means of getting to the ground. This would stop this particular engine from being used for anything other than space flight regularly. There could be another engine which is essentially an atmospheric engine that could be used for space travel which airliners and/or flying taxis could use but it would effectively be the reverse idea of this engine.
With regard to fuel technology etc. I would suggest that your civilization had a leap forwards in its energy storage technologies, allowing for massive amounts of power to be stored in rather small devices and that the engine generates its thrust from some electrical method, this way solar panels can allow for collection of energy whilst in transit. The only other way is that maybe whatever fuel the ship uses could be skimmed whilst orbiting a star or some kind of planet but the energy output of such a material would have to be enormous as the best description for our rockets I have heard goes something like this:
We put three men on top of some of the most explosive substances known to man, we point it upwards, light the fuse and see what happens.
So anything that provides enough fuel to get up and down without refueling needs to be much more powerful than what we have today, I would suggest that such a material is highly explosive (think the power of the sun in a can) but only under the right circumstances otherwise a civilization could wipe itself out with one small accident and wars could be won by just blowing up one small installation.
As for the look of the craft, it could be as varied as you like in the instance of a slow descent, it just needs to be strong enough to support its on weight in atmosphere and keep the atmosphere inside when in space.
In the instance of using the atmosphere to slow down, it would probably need to be aerodynamic in some way with the ability to become less so at will. If you want it to take off like a plane then it would need to be roughly plane shaped but maybe with the option of deploying some kind of flaps that connect the wings and rudder to increase aerodynamic drag and slow down the ship more efficiently during landing. If you want it to take off like a rocket then it would need to be roughly rocket shaped and maybe the front section can be pulled in towards the body, splaying out the nose to create kind of an umbrella effect with the nose, again, increasing surface area during landing.