I will split this answer in two halves, for two different kinds of date expression.
On November 4
The traditional Roman calendar, whose system is still in use in some festive occasions, is based on three special days in each month:
Kalendae (first day of the month), Nonae (fifth or seventh), and Idus (13th or 15th).
The later options (7 and 15 instead of 5 and 13) were used in March, May, July, and October; in the other months Nonae was fifth and Idus thirteenth.
All these three words are plural feminines.
The name Nonae should not be confused with the feminine plural of "ninth" (nonae).
Capitalization and inflection both hint which one is meant, as feminine adjectives are not needed for dates.
There are "ninth" days in the Roman calendar, but they occur in connection with Kalendae, never together with Nonae.
If the date falls on one of these special days, you simply put it in ablative:
- on March 1 = Kalendis Martiis
- on August 5 = Nonis Augustis
- on October 15 = Idibus Octobribus
In case of days immediately before special days, use the word pridies in ablative and put the special day in accusative.
The name of the month is that of the special day, not necessarily that of the day of interest.
- on February 28 (29 on leap years) = pridie Kalendas Martias
(I have sometimes seen postridie used the same way for days right after a special day.
If this method is used, then "on August 6" = postridie Nonas Augustas.)
For all other days, you count days to the following special day.
As is typical in Latin, counting is inclusive:
- on October 13 = ante diem tertium Idus Octobres
- on October 12 = ante diem quartum Idus Octobres
- on October 17 = ante diem sextum decimum Kalendas Novembres
All these can be abbreviated, of course:
- Kal. Nov.
- prid. Id. Aug.
- a. d. XIV Kal. Dec.
Counting days before Kalendae Martiae on a leap year is a bit tricky, and I will exclude it here.
This special case and the story behind it should be taken to a separate question.
In contemporary Latin, it is common to write things in a more simple and modern way: "on January 5" is simply die quinto mensis Ianuarii.
This can be abbreviated further without much risk of confusion.
The first option I propose is to leave out die and mensis, since they can be understood implicitly, leaving us with quinto Ianuarii.
Another option is to take the month name (which is an adjective in Latin) to modify the numeral or the implicit dies, leading to quinto Ianuario.
A year can be added by the ablative of annus and an ordinal numeral: Hic situs palam factus est Idibus Martiis anno bis millesimo sexto decimo.
Today is November 4
I am not aware of any ancient use of phrases like "today is November 4".
In fact, pridie is often considered an adverb since the noun pridies is only found in the singular ablative form used in dates.
However, the old system can be adapted to such expressions:
- today is August 5 = hodie sunt Nonae Augustae
- today is March 31 = hodie est pridies Kalendas Apriles
- today is October 12 = hodie est dies quartus ante Idus Octobres
I stress that these are not attested (as far as I know), but I consider them to be a reasonable extension of classical Latin.
Notice that hodie is here an adverb, not the subject in the above sentences.
Using the more modern variants, the construction is simple: "today is August 5" = hodie est dies quintus mensis Augusti.
Again, ellipsis is possible if this feels too verbose.
A year can be added like in the "on November 4" case.
But now genitive is more appropriate than ablative: "Today is December 6 of the year 1917." = Hodie est dies octavus ante Idus Decembres anni millesimi nongentesimi septimi decimi.
Let me end with listing all the possibilities I offer for November 4.
On November 4:
- pridie Nonas Novembres (prid. Non. Nov.)
- die quarto mensis Novembris (die 4 mensis Nov.)
- quarto Novembris (4 Nov.)
- [die] quarto Novembri (4 Nov.)
Today is November 4:
- hodie est pridies Nonas Novembres
- hodie est dies quartus mensis Novembris
- hodie est quartus Novembris
- hodie est [dies] quartus November