Technically there is no end to how good a photographer you want to become. But learning basic DSLR camera controls and options takes atleast 1-2 days. Courses start from 1 day and go up to a few weeks. You have to basically learn to reduce the usage of Automatic mode and instead start using the Manual modes (M, A, P etc). You keep improving with practice and can reach expert levels in a few weeks.
A good photography course covers how to adjust the Camera controls against several important aspects like Aperture, Shutter Speed, Focussing, Depth of field, Exposure, Metering, Flash etc.
Once you learn the controls, it needs a lot of practice in different conditions of Ambient Light, Size of subject, Movement of subject, Foreground vs Background, Distance of subject, Colours etc. It is followed by some level of optional processing to enhance the effect. You need to have a fair idea of working with photo processing software like Adobe Photoshop, Paintshop, Corel Draw.
To learn different specialised styles of photography like Macro/Close-up (photographing minute subjects), Landscapes/Zoom (faraway objects), Portraits, Sports, Night photography etc you will need to learn what type of lenses and settings to use. Lenses is a big subject in itself. There are lenses for every imaginable purpose.
For learning professional photography, there are several free courses available online. There are even Degree courses. Manufacturers like Nikon and Canon have their own free learning programs. But for a novice, it is hard to understand and follow all those instructions. It becomes confusing after a while if you depend only on self-study.
It is always better to learn from an experienced practioner cum trainer in a classroom. Most of them follow up the classroom sessions with field trips because you learn a lot of variations on the job. Atleast learn the basics in a classroom and do your research on advanced photography aspects.
If you want to become a professional photographer, first identify what type of photographer you want to become - Portraits/Wedding & Celebrations/Wildlife/Sports/News/Ads/Films etc and then take up the course accordingly.
Remember that having a most advanced and expensive camera is not the criteria for good photography. Some people take better photos with a regular digital point & shoot camera than another person with a high-end DSLR. That means you need to have the eye for detail, aesthetic sense, timing and taste along with good equipment.