One little goat, one little goat that my father bought for two coins 
One little goat, one little goat

The following paragraph is a different interpretation as to what the first stanza is referring to.

(The goat is the firstborn that Yaakov bought from Esav with bread and lentils, which correlates to “my father bought for two coins”, and then gave it over to Yosef. In fact, the small numerical value of Bechora is the same as the numerical value of Gedi [goat]. Subsequently, when the brothers sold Yosef, he was 17, which is the same numerical value as Gedi. That is what it means when it says the cat ate the goat.)

All the good that Hashem bestows upon us is simply inherited from our forefathers. Were it not for Yitzhak’s blessing to Yaakov, everything would have gone to Esav and we would have been left with nothing. What was the source of this blessing exactly? According to the Gemara it was the two goats that Yaakov brought to Yitzhak on Erev Pesach to receive the blessing. As the sages teach, they are ‘good for you and for your children’, meaning one is good in that he would receive the blessings and the other is good for his children that they will bring two Se’irim to atone for their sins on Yom Kippur. The Gr”a explains that the animals that Yaakov brought and the Se’irim actually worked the same. Just like the two Se’irim were brought for atonement and to appease the Satan, so too the two goats that Yaakov brought were to receive the blessings and to appease the Satan.

Yaakov then proceeded to give these blessings over to Yosef. Now one may question, didn’t he bless all of his children? The Gr”a points out that regarding each of the other tribes, the Torah does not use the word “blessing”. Only with regards to Yosef do we see it actually described as blessings.

Along came a cat and ate the goat 
Along came a dog and bit the cat that ate the goat 

However, the brothers got jealous, which is the nature of a cat, and sold Yosef, who represents the goat, in order so that they could inherit his blessings.

After Yosef’s ultimate passing, Pharaoh, who is compared to a dog, put the tribes into slavery as punishment for the sale of Yosef.

Along came a stick and beat the dog
Along came a fire and burnt the stick

The sages write that when Moshe arrived to save the Jewish people and remove Pharaoh from power, he hit Pharaoh as if he were a dog with his famous staff.

After the redemption, the entry into Eretz Yisrael, and the building of the Beit Hamikdash, the fire came along to destroy all they had ever worked for. The Yetzer Hara of idol worship enticed the Jewish people to reverse what the stick represented, which was redemption, and led to the burning and destruction of the Beit Hamikdash.

Along came some water and extinguished the fire

Nevertheless, since the Jewish people are impervious to destruction, they put out the fire. Water, which the Jewish people are compared to, extinguished the fire of the Yetzer Hara. According to another interpretation, the water represents the Anshei Knesset HaGedola that rebuilt the Jewish nation.

Along came an ox and drank the water

Unfortunately, Edom, a descendant of Esav, who is referred to as an ox, destroyed the Beit Hamikdash and propelled the Jewish people into the Roman exile that we are still suffering through today. As a side note, the first destruction is compared to fire because the main sin was the fiery idol worship. On the other hand, the second destruction is compared to an ox since our baseless hatred brought it with the strength of Esav, who is compared to an ox.

Along came a slaughterer and slaughtered the ox

But there is always hope. Mashiach ben Yosef, will come and save us from this 2,000-year exile. In fact, it is very fitting since Yosef is also referred to as an ox.

Along came the angel of death and killed the slaughterer

Along came the Holy One blessed be He and killed the angel of death

However, as is recorded in the Seforim, Mashiach ben Yosef will be killed. Nevertheless, Mashiach ben David will be the ultimate savior, as when the hand of Israel rises up, the blessings return. Every time we lose something, we get everything back. Every time we do teshuva, we get everything back. Therefore, we return to the beginning; One goat…