Eliminate the unnecessary thinking process, and go for the modified wineglass grip that is well established within the community. It is the most common grip for a reason. It is when the joystick goes between your pinky and ring fingers and your thumb rests up top. It works fantastically well for both beginners and intermediates alike. Many who have gotten used to unorthodox grips wish they would have started with the modified wineglass grip from the get go.
While you could definitely start practicing with your main character straight away, your main character might be too advanced for your transition. Thus, it is not wrong to switch to an easier character as you transition your way to the glorious realm of the arcade stick.
Utilize training mode to practice normal attacks and special moves on both sides of the screen. Remember to turn on input display if supported by your game of choice. As you get more comfortable, you can start attempting combos. Aim to reach the level that you were at with a regular controller. keyboard, or whichever inferior apparatus you were using prior. Once you become comfortable enough, activate the dummy so that you can fight it as CPU.
First, start with the lowest difficulty and then ramp it all the way up to the highest difficulty. Once you reach that point, either play the CPU in versus matches or start playing with your sparring partner after giving him or her the disclaimer that you are still getting used to playing on an arcade stick. Optimally, he or she would be in your same position as well, trying to get the hang of the craft like you are. After that, take the arcade stick online to play some friendly matches, eventually making your way to ranked matches. Your final stop should be the tournament stage!
Although there is no universal answer to the question of when to finally make the switch to an arcade stick and drop the controller completely, fighting gamers usually find themselves making the switch as soon as they find themselves reaching a higher win-loss ratio than when they played with a controller despite the fact that their execution might still not be on par, such as struggling on one side of the screen more than the other.