You can learn to drive by taking classes at the Kuwait Motoring Company. They have a branch in Shuwaikh next to the MOI vehicle inspection center. They may have other branches too but I'm not sure about that, better to contact them and ask. The advantage with them is that their campus is in the same place where you get to take the drivers test, so you'll be testing at the same track as the one you practiced with.
Though I don't recommend learning to drive here if it's your first time... the driving culture in Kuwait is very poor; most people people drive aggressively with no consideration of others on the road, a general lack of driver ethics and awareness, as such this leads to a lot of people getting into accidents on the road. You'll be picking up a lot of bad habits in this environment.