Dating the Festivals of God:

(an excerpt from ‘The Passover and The Messiah' – link here )

Each biblical month is a lunar month of some 29 or 30 days and begins when the New Moon is seen, or more accurately on the night (the beginning of a biblical day, as its starts at sunset) that the first crescent sliver is seen after the days/nights of the New Moon.

The New Moon is the opposite of the Full Moon, that is, it occurs when there is no sunlight reflecting off the moon’s surface because the moon is between the sun and the earth. 
This lasts for between 1.5 – 3.5 days every lunar phase and around 2.5 days in the Middle East. 

Thus it would be difficult to know which of these 2 to 3 days to start the month on, if you based the start on the New Moon (when the moon may not even be visible – see image below) itself rather, than the first sliver of light or crescent reflection after the New Moon.

As in ancient times the people worked from dawn to dusk, they would have observed that the moon was still a ‘new moon’ when they started work on the 2nd  or 3rd  day and then seen the beginnings of the bright crescent caused by the sunlight reflecting off the moon’s surface that night as dusk arrived and they finished work (if there was cloud cover, this was made a little difficult).

There is also no sunlight reflecting off the moon when we have a total lunar eclipse, but this is quite a rare and separate event.

Thus Hodesh, the Hebrew word for month, while it literally means ‘New Moon’; actually refers to when the first crescent sliver appears as just explained.

Ps 104:19 "He created the moon for Mo'adim [appointed times]"
Or in the NASB “He made the moon for the seasons;"Lev 23:2: "These are the Mo'adim [appointed times] of YHWH, holy convocations which you shall proclaim in their appointed times [Mo'adam]."Or in the NASB: "Speak to the sons of Israel and say to them, 'The LORD'S appointed times which you shall proclaim as holy convocations-­‐-­‐My appointed times are these:”

These passages show us that it was God’s plan for us to use the moon (or moon’s phases) to determine the times of His special days.

In 1 Sam 20:5 David says to Jonathan "Tomorrow is the New Moon (Hodesh)". Here you can see that while the Hebrew word 'Hodesh' is taken to mean month, strictly speaking it refers to the first day of the month, the ‘New Moon’ or the first crescent sliver.
Look at Ex 13:

“1 Then the LORD spoke to Moses, saying, 2 “Consecrate to Me all the firstborn, whatever opens the womb among the children of Israel, both of man and beast; it is Mine.” 3 And Moses said to the people: “Remember this day in which you went out of Egypt, out of the house of bondage; for by strength of hand the LORD brought you out of this place. No leavened bread shall be eaten. 4 On this day you are going out, in the month Aviv.”

As we shall see, God changed this month’s name to Nisan (meaning first or first fruits). But what does Aviv mean?

Aviv (or Abib)  indicates a stage in the development and ripening of the barley crops. This stage is indicated in Ex 9:31-32 which describes the devastation caused by the plague of hail:
"And the flax and the barley were smitten, because the barley was Abib and the flax was Giv'ol. And the wheat and the spelt were not smitten because they were dark (Afilot)."

Being Aviv meant that the barley had reached a stage of its ripening where it was brittle and therefore more susceptible to damage from the hail.

Barley which is in the state of Abib has 3 characteristics: 

1) It is brittle enough to be destroyed by hail and has begun to lighten in colour (it is not "dark"). 

2) The seeds have produced enough dry material so it can be eaten parched. 

3) It has developed enough so that it will be harvest-ready 2-3 weeks later. 

See http://www.karaite-korner.org/abib.shtml

Most versions translate this poorly and therefore miss this connection.

Surprisingly, the NIV is helpful here: Ex 9: “31 The flax and barley were destroyed, since the barley had headed and the flax was in bloom.
32 The wheat and spelt, however, were not destroyed, because they ripen later.”

From this it is clear that the barley, which was Aviv at the beginning of the month, has become harvest-­‐ready 15-­21 days later (i.e ready to be used in the ‘first-fruits’ or wave-sheaf offering on the first day after Sabbath during Passover/Pesach).
Therefore, the month of the Aviv cannot begin unless the barley has reached a stage where it will be harvest‐ready 2-3 weeks later.

That the barley must be harvest-­‐ready 2-3 weeks into the month of the Aviv is also clear from Deut 16:9 which states: "From when the sickle commences on the standing grain you will begin to count seven weeks.”

 (This passage is referring to celebration of Shavuot, the Festival of Weeks – also known as Pentecost.)

As a result sometimes (as in 2006), Barley was not ready, that is, was not Aviv in Jerusalem, and so another month called VEADAR was added (the 13th month of the Biblical year and hence it was a biblical leap year) and this resulted in Passover being a month after Easter.

What is the practical impact of this today?

In part it means that wherever you live you can ascertain the start of the month by observing when the New Moon occurs and then waiting 2-3 days until you see the first crescent. This will be the start of the month.

Update:
So it's now September 2017 and the New Moon and brings in Yom Teruah (Rosh HaShanah/The Day of Shouting - the blowing of the Shofar) is expected on the Friday 22nd September 2017, but you can verify it for yourself.

And what makes Yom Teruah an especially exciting and hopeful time?

Please see my articles here (The Day of Trumpets) and here  (10 Questions on Yom Teruah).

Note: 

Even though I would argue that the calculation of the start of the new (Biblical) month (from observing the first crescent after the New Moon) is totally valid and most Biblical, it would seem just as important and perhaps even more commendable to celebrate the Festivals of God from Peasch/Passover to Yom Teruah and Yom Kippur in solidarity with the State of Israel, and therefore on the one (or two days outside of Israel), that the Jewish people in Israel are celebrating it on.

You can find the dates that the State of Israel observes at either Aish.com or Chabad.org