Is it possible to turn gas giants like Saturn or Jupiter into planets with solid ground and desired atmospheric composition?
Yes and no. The main problems will be surface gravity, temperature and irradiance.
With the energy and resources required to pseudoTerraform a gas giant, you could probably tear it apart, discard everything except the material required to build another Earth, and do so.
The first step is to transform a sizeable portion of the planet into some composites with very high compressive strength and very low density. With this, you expand the planet until its surface gravity is around 1G. You need to have enough material left to cover the results with a thick rocky crust.
In the case of Saturn this is actually not so inconceivable, because its "surface" gravity is already close to that of Earth. You might "only" need to dredge up enough rock to build the new surface and the structures required to make it buoyant above the remaining atmosphere. Carbon and silicon aerogels might do. So you end up with about 80% of the old planet, plus a layer of churned waste materials, covered by a thick aerogel crust, covered by rock, soil, and finally Earth-like atmosphere.
Except that at this point you need some heat source, because at Saturn's distance you're going to experience quite a chilly climate, and greenhouse effect can only do so much.
You could use a lot of the hydrogen as energy source for fusion plants, and use humongous helium ion engines to slow down the planet and bring it closer to the Sun. It would probably require thousands of years, though.