Although there many who recommend manual for learning, I think most of the time it's much better to use one of the semi auto modes & just set the parameter that is most important to your shot.
For these sort of shots I'd be inclined to use Av mode setting the aperture for the DOF you want. Then leave the camera to sort out the right speed to go with that. This should allow the exposures to be acceptable without needing high ISO, and will also limit the amount you have to learn to get good shots.
I probably work in Av 80% of the time, switching to Tv (time value or shutter priority) when shutter speed is more critical than DOF. Manual only gets used when I need full control, or light levels are too dark for the cameras meter.
Getting the camera focused where you want may be another part of the problem if the images are soft as well as grainy.