Created a field with the formula Today + 2 days, checked the next day, the date is old, i.e. in which this column was created, I need the date to be updated every day, how to implement it?
If you create a Today column it needs to be updated. You can do that with either a timer job or by placing a jquery script on a page that is hit by the user. The script could call SPServices.SPUpdateMultipleListItems to do the update. Pass a CAML clause so that you only update the list items where the Today value needs to be updated, e.g. once per day.
Formulas in calculated columns are only recalculated when the item is saved. The value is static until it is saved again. So the value for [Today] will be the date the last time the item was updated and won't change again until the next time it is updated.
One option might be to write a PoweShell script to read and update every row in the list and run it on the scheduler every day.