If you're talking about rockets that are currently out there, the only ones that would really be suitable are liquid or hybrid chemical rockets. Liquid chemical rockets use fuel and oxidizer that is in liquid form. A hybrid rocket has typically a fuel that is solid and an oxidizer that is liquid. Solid chemical rockets also work in and out of atmosphere but aren't really throttleable.

Ion engines, as you said, produce too little thrust to be useful in atmosphere and even require different mission planning in space than chemical rockets due to their incredibly low thrust. It's conceivable that with enough electric power, you could make ion engines produce thrust closer to chemical rockets, but we're talking about obscene levels of power required to achieve this, which would currently be in the realm of sci-fi.

Nuclear rockets are also a thing that have been tested and use the heat of a nuclear fission reaction to accelerate a fuel such as liquid hydrogen to produce thrust. Due to the low molecular mass of hydrogen, they produce less thrust than typical chemical rockets but are more efficient than chemical rockets and are also throttleable.