The month() function from the lubridate package will extract the month value from a date. If your data frame is named DF and the column is named Date, you can make a new column named month with
library(lubridate)
DF$month <- month(DF$Date)
That will work if the Date column is a numeric date. If the Date column is characters, it should work to do
DF$month <- month(as.Date(DF$Date))
Once you have the month column, you can sort on that.