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I want to have an Earth like planet, but with most of the continents having very little flatland. Thus continents would be covered mostly by hills and mountains. I want the rest of the planet to be as similar to Earth as possible in size, gravity, ocean coverage, atmosphere, etc.
How do I explain this?
That sort of terrain might come about as a result of this planet having many more tectonic plates than Earth.
For example, when the planet formed, these plates may have collided much more chaotically, and violently than Earth's, thus resulting in more mountain ranges than we have.
This will also result in a lot more volcanic and earthquake activity than Earth. These in turn can also shake the landscape up.
Add to this some meteor impacts in the distant past, and you're probably covered.
Give it smaller continental plates and speed up their movement. Mountains are raised where one plate pushes into another.
A lot of edges gives you a lot of territory that mountains are being created in.
A recent Snowball Earth period might help.
Heavy glaciation can cause enormous changes to the terrain. There are no longer any glaciers where I live, but there once were, during a recent ice age. Their legacy includes
The formation of glacial valleys in particular can be quite extreme:
Gif courtesy of Wikipedia user Intexpliki under the Creative Commons Attribution-Share Alike 4.0 International license.
You could go with a normal ice age, but you won't see glaciation over all of the planet, just portions extending out from the poles, perhaps into the southern temperate zones. That's why a Snowball Earth scenario is better.
Make your planet younger than earth. Younger planets are more geologically active and have had less erosion - they will naturally be more mountainous/hilly. Though with some other more troublesome attributes to carefully manage in your story!
Evolution of Terrestrial Planet Surfaces
The link above discusses the effects of impact cratering, tectonic activity and then erosion during the evolution of a terrestrial planetary body's life cycle. In short - terrain will tend from geologically active and "rougher" to smoother less geologically active as a planet ages.
Well, I'm going to add an option: crust-buster bombs.
That's right, ladies and gentlemen, we've created weapons capable of simultaneously creating a hole from the surface to the mantle and the pressure waves necessary to trigger an instantaneous volcanic eruption through that same hole! Thanks to our new handwavium technology, this awesome force can be yours for the mere price of... (Terms and Conditions may apply, not guaranteed to work in areas with thicker-than-average crust such as existing hills and mountains.)
I'm sure you can find some sort of way to justify a world where people went crazy with the crust-busters and basically turned the entire planet into a series of volcanoes for a few months. Bam!!! No more flatlands.
If you want a more 'natural' explanation, I can't do better than Andrei.
