I have not found that any of the commentators discuss this point, so I will attempt to make my own analysis.

1) First, I am assuming that your question relates to the Gemara from 56b onwards, where the Gemara starts to teach what different dreams represent. In other words, the matter at hand is to try and understand the Gemara that starts with Rebbi Chanina's statement that someone who sees a well in his dream will experience peace. I will try to explain why the dream about the well is the first one for which the Gemara provides us with a general explanation, in contrast to the dreams of individuals which had been mentioned up to now, for which the interpretation of various Sages had been given.

2) We should note at the outset that, in principle, dreams are good things. This can be learned from the Gemara earlier (55b) which teaches that it is not a good thing to go seven days without having a dream. Therefore, when the Gemara commences to explain different dreams, it starts with good dreams and teaches us what the best kind of dreams are.

3) This is why Rebbi Chanina explained first the dream which symbolizes peace, because peace is one of the greatest blessings of the Torah. This can be seen from the very last passage in Maseches Berachos, which cites the verse (Tehilim 29:11), "Hash-m will bless His people with peace." To stress further the tremendous value of Shalom, it is not only the first tractate of the Talmud which concludes on the note of peace, but in fact the last tractate in Shas also concludes with Shalom. The last Mishnah in Maseches Uktzin, at the very end of Seder Taharos, states, "The Holy One, blessed be He, did not find a vessel which contains blessing for Yisrael other than peace, as is said... Hash-m will bless His people with peace."

4) However, the Gemara proceeds to show to us that there is another blessing which is just as great as peace. This is the blessing of Torah. This is also stated in the words of the aforementioned verse, Tehilim 29:11. Immediately prior to being told that Hash-m will bless His people with peace, we are told that Hash-m will give "Oz," strength, to His people. The Gemara in Zevachim (116a) tells us that Hash-m said that He possesses a great delight which had been hidden away in His treasure house for 974 generations before the creation of the world, which He desired to give to His children. Everyone proclaimed, "Hash-m will give 'Oz' to his people." Rashi explains that "Oz" refers to the Torah, which is the "strength" of Yisrael.

5) We learn from this that the blessing of Shalom and the blessing of Torah go hand in hand, and are two of the greatest blessings that Klal Yisrael can receive. This is why -- immediately it cites Rebbi Chanina -- the Gemara proceeds to cite Rebbi Nasan who says that the dream about the well signifies that the person will find Torah.

6) Above, I noted that dreams are essentially a good phenomenon, so it is logical that the Gemara should tell us about the best kinds of dreams first. We saw that this is why the Gemara commences with the dream about the well, which is an omen for the great blessings of peace and of Torah. We now observe that the Gemara mentions another interpretation of the dream of the well. This is the interpretation of Rava, that the person who dreams about the well will find life itself.

7) The blessing of life is the greatest blessing possible. Our Sages say this in Kidushin (80b) in their explanation of the verse (Eichah 3:39), "What can a living person complain about?" The Gemara explains that a person can never really have any complaints, because "it is sufficient that I have given him life."

Life itself is the greatest gift that we ever received from Hash-m. Every moment of life is priceless. This is the reason why the Gemara mentions the dream of the well as the first of the general dreams; it signifies the three greatest blessings in the world: Shalom, Torah, and life itself.

8) I will not attempt to trace the connection between all of the dreams mentioned in the ensuing Gemara, but I believe that if one looks closely one will observe a logical sequence. Nonetheless, I will cite one of the commentators, who -- although he does not relate directly to the order of the Gemara -- touches upon the depth of the dreams discussed in the Gemara.

9) The Gemara (57a) mentions dreams about wheat and about barley. The Gemara first tells us that if one sees wheat in his dream, he will see peace. Then the Gemara says that if one sees barley in his dream, this signifies that his transgressions have disappeared.

10) I saw a beautiful explanation, in the name of Chasam Sofer, about the significance of the dreams about wheat and barley. They represent Shavuos and Pesach, respectively. On Pesach, we bring the Korban ha'Omer from barley. On Shavuos we bring the Korban Shtei ha'Lechem, two loaves of bread made from wheat. We may add that there is a hint here that the festival of Pesach represents turning away from sin, while the festival of Shavuos represents a higher level, where we attain Shalom, since we received the Torah on Shavuos, and, as we mentioned above, Torah and Shalom go hand in hand.

11) The significance of Pesach representing forgiveness for our sins is that on Pesach we are very careful not to eat even the smallest amount of Chametz, because Chametz represents the evil impulse, which we must attempt to eradicate. Once we have done this we can reach a higher level of actively doing good, which we attain through the Torah that Klal Yisrael received on Shavous.

12) This is just one example of the inner meaning of the dreams mentioned in our Sugya, and if would examine the other dreams, we would no doubt also find deep explanations.

Dovid Bloom

I posed your question to a Gadol and he replied that it is not known whether or not there is any order to the dreams mentioned in the Gemara.

Dovid Bloom