Others have pointed out that, even given some miracle power source, the energy and momentum of the rocket exhaust (as need to lift off) will be quite destructive.
My variation of that is: what if the rocket exhaust was not? Spray high mass, fast-moving particles about in great abundance, and have it not affect the surroundings.
I think the general idea as used in a Robert L Forward short story used neutrinos.
But you would do better to make a dark matter rocket since dark matter particles (specifically WIMPs) are heavy. This lets you use less energy to get the needed momentum (at the expense of consuming reaction mass).
So, what you need is a rocket that then changes the exhaust material into dark matter after you have manipulated it. That’s how neutrino beams are made: you form the beam with charged particles so you can manipulate them, and then those decay into neutrinos.
A dark matter rocket would seem to the outside to be a reactionless drive. It could be in the middle of the ship without an exhaust port.
speculative fiction mechanism
What kind of techobabble could do this? “supersymmetry”. And as far as we understand (cold dark matter being a thermal relic of the big bang) this is a real thing! Some as yet unknown mechanisms can “flip” matter into dark matter at very high temperatures.
Then, rather than just being hot in a thermal mess, you carefully tune the reaction by using “resonances” to set up the exact energy needed. This makes more efficient conversion without waste. This is the same thing you can bring up to make the fusion power plant work! So, explain that the same tech that allowed fusion rockets to be built also (once the details of the particles were known) to turn that into a dark matter rocket.
Looking up some links to explain how the “thermal relic” implies these things, I thought of another way. This also unifies with the power generation.
The ability to convert matter (as happened in the early universe) depends on multiple R-parity violations (that is, more than one process) that occurred at extraodinary temperatures. Now, to get it to happen at conditions possible to produce today, rather than cooking it at some ridiculously high temperature (so even rare things happen) you vary carefully tune the energies and reactants. Like a combination lock, multiple pathways must be promoted (by providing narrowly tuned conditions) at the same time. Note that careful tuning vs outrageous temperature is the same idea as before. But here, invoking “multiple channels at the same time” gives us the combination lock analogy.
Now here’s the kicker: the same stuff can induce proton decay. Make that part of the plot: not only do we have a power source from ordinary matter, that’s the very process that can produce dark matter beams.
JBiggs’ answer, stripped to the essentials, is that given enough power (e.g. nuclear fusion) an airplane would be fine. Airplanes (VTOL, helicopters, and whatnot) fill the OP’s requirements on unobtrusive takeoffs.
What you need, then, is to accelerate to 18,000 miles per hour. With enough raw power, a jet can do that, using the atmosphere. However, look at the real-world reentry to see the problem: with thick air the ram pressure will bake the craft even if you had power to push through the air. With thinner air, you need a big scoop to gather enough air to attain sufficient thrust.
So suppose in airplane mode it simply flies as high and as fast as possible; that’s still a long way from orbital speed which must be achieved outside of the atmosphere. So you must switch to rocket mode. But, you don’t require the absolute thrust needed to lift vertically off the Earth! You could do with a low thrust, focusing on efficiency of reaction mass instead. Say you start at 2000 mph and angle your jet to gain altitude until the air is too thin to work; then fire the rocket at a fraction of a g thrust, building speed. A few minutes later your arc takes you back down to the air and you arrange to “skip” off the air! Repeat with successively longer hops as you gain speed and attain full orbit.
After writing that, I recall hearing the concept for real. Not for reaching orbit, but for an extended flight at mach 10 using air-breathing engines. I managed to find the reference to Hypersoar by Lawrence Livermore National Laboratory.
Why not look into what real-world rocket scientists think of the idea of single stage to orbit like the Skylon or other spaceplanes. Look over near-future designs, and figure what you might do with sufficient power and how light you would need to build beyond current materials, and design your craft based on that.