To handle the month information you could do something like:
Month <- factor(Month, levels = month.abb)
This will convert the Month column to a sorted factor. month.abb is a built in vector of abbreviated month names. For full month names, use month.name.
You could use a similar approach on the Day names, sorting according to your preference i.e. week starting on Sunday or Monday. I don't know of a built in vector of day name, but it's probably better to create your own ordering.