Washing your white shirt is better than get it dry-cleaned. You can keep look of shirt ultra white by washing it with detergent specially formulated for white clothes. Before washing carefully read the washing instruction on the label of the shirt. These instructions guide you regarding washing requirements for the specific type of cloth. You should separately wash white clothes from colored clothes.
To make bright look of dull shirt you can use fabric bleach specially formulated for white clothes. If you go with normal fabric bleach it can make your shirt yellow. You can always use stain remover to remove dust and stains from cuffs and collars of your shirt. Use washing brush to apply stain remover on cuffs and collars then rub it smoothly because while rubbing hard it can damage the fabric of shirt.
If you are facing problem of underarm stains which are due to deodorant or marks of perspiration, then you can sew patch guard designed for garments. After washing you should dry your shirt on hanger because it minimizes its creasing. In wardrobe use hanger of wood rather than wire as it keeps the shape shirt for long time. Do not hang white shirt with color one because it can rub off from other one.