In mainstream Sunni teaching, the authentic practice for Ashura is fasting. Remembering Allah and giving charity are good in every month, but no special ritual for this day was reported. Many other practices are attributed to this day, such as special prayers, special foods or adornment, but no authentic report supports them. Mourning rituals, and also celebrations or festivities, were not taught by the Prophet ﷺ for this day, so they are not part of the authentic practice. Muslims are encouraged to keep to what was reported from him.
This is said with respect. Muharram is a month in which many Muslims remember painful events in history, and all Muslims honor the family of the Prophet ﷺ. Different communities have their own traditions, and this article only describes what the authentic reports establish. Please ask a qualified scholar if you have questions about your own situation.