We have quite a number of episodes to explore, so let's start organizing. Again, there are some episodes I can safely place before others in the chronology, and in a particular season, but a lot of this is based on similarities to our world, not necessarily explicitly stated in the show. If you see anything I've said that doesn't match with something explicitly stated, please mention in the comments and I'll add an addendum at the end to reflect the changes in the timeline.
The Grand Galloping Gala Arc of episodes ties together a large number of episodes from the first season into a single cohesive story. These episodes are spread pretty throughout, with the final episode ending with the gala itself. The episodes in question? The Ticket Master (has to be first, due to the fact that the Gala was first brought up there to begin with), followed by Suited for Success (the dresses have to be made for the gala in this episode), Green Isn't Your Color (Fluttershy's dress made in the previous episode appears on the dress rack in the episode), and of course, The Best Night Ever, the episode with the Gala itself. Later on in season two, in Sweet and Elite, the Gala is mentioned as "coming up soon" by Twilight in response to Rarity's co-mingling. This places this episode somewhere in between Ticket Master and The Best Night Ever, but it's difficult to say exactly where. Due to the placement of the episodes, it's safe to say that this all takes place within the first summer of the first year.
The Cutie Mark Crusaders are an important set of characters in helping us define what happens when. Due to the settings in each episode starring them, the fact that they all attend school, and if we are to assume that school, like ours, starts in the late summer/early fall, then we have a good idea of when every episode starring them occurs throughout the year. As the CMC first met in Call of the Cutie, we can assume that episode came first throughout the CMC series of episodes. Every episode following that is a bit iffy in placement, but we can place a few in order. For instance, Babs was introduced for the first time in One Bad Apple, and brought up again in Apple Family Reunion, making those two episodes happen in that order. However, due to the large number of episodes containing them, it's neigh impossible to ascertain the proper order. In general, though, I believe all of these episodes fall into the late summer/autumn category of the first year (any episodes between season one and season three) and into the late summer/autumn category of the second year (season four).
Owlowiscious is first brought up in Owl's Well That Ends Well, followed by May the Best Pet Win! (which introduces Tank to the cast), Magic Duel, Just for Sidekicks/Games Ponies Play, Maud Pie (in a second pet playdate), Testing Testing 1, 2, 3, Inspiration Manifestation, and Twilight's Kingdom Part 2. Because the pets tended to be placed in the same episodes together, Owl's Well That Ends Well preceding May the Best Pet Win!, followed by some order of the other episodes would be a safe conclusion to reach. Because of the arrival of Tank, we can place May the Best Pet Win! before the following episodes (again, I can't guarantee the order): Too Many Pinkie Pies, Just for Sidekicks/Games Ponies Play, Magical Mystery Cure, Pinkie Pride, Maud Pie, and Testing Testing 1, 2, 3. Of course, despite all of these, because of the presence (or lack thereof) of Twilight's alicornication, we know that the season four episodes had to take place after the others. I won't go into each individual pet (Angel has 24 episode appearances by himself), but data extraction should be about the same for the other pets.
This one includes various episodes from season three and season four, but as compared the previous sections, we can safely say that most of these happen within the span of the second year (after the second Summer Sun Celebration, or the start of season four). In season three, we have four episodes to place: The Crystal Empire Parts 1 and 2, as well as Just for Sidekicks/Games Ponies Play. The Crystal Empire had to return before they could host the games, followed by the Mane Six pushing for them to be accepted as the Games location that year in Games Ponies Play. This had to have happened in the spring right before season four began, given how much goes into preparing for the games throughout the fourth season. Other episodes that feature the games (prominently) are Flight to the Finish, Rainbow Falls, and, of course, Equestria Games. Flight to the Finish and Rainbow Falls happen in some order, but everything else retains the same order that they were broadcasted in. More than likely, the Equestria Games happen during the late summer/autumn of the second year.
This one was one that threw me for a loop initially, but proved even more so than any other piece of evidence that the episodes are broadcasted out of their chronological order. In Twilight's Kingdom Part 1, Discord is shown with the book in which they recorded the key events throughout season four. Given that the book started empty and ended (relatively) full, we can safely assume they wrote in the book like any normal pony would: from left to right, from beginning to end. When Twilight begins reading the journal, we're shown where the placement of the colored bookmarks are in the book, each corresponding to one of the Mane Six. The order shown? Orange, yellow, white, blue, and pink, corresponding to Applejack, Fluttershy, Rarity, Rainbow Dash, and finally Pinkie Pie. Because of the journal, we know that the episodes had to happen in this order: Leap of Faith, It Ain't Easy Being Breezies, Rarity Takes Manehattan, Rainbow Falls, Pinkie Pride, and finally Twilight's Kingdom Parts 1 and 2. This is especially important, given that Leap of Faith, in broadcast order, was the last (besides the finale) to be shown, despite chronologically being first.
Season four caused me quite a bit of duress, simply because no matter how I looked at it, a large number of the episodes had to have happened in the spring of the second year. The episode? It Ain't Easy Being Breezies. As they're collecting pollen from the flowers, it's necessary for it to happen in the spring, and given the chronology stated above, every episode listed had to also happen in that spring. Some CMC episodes (like listed above) had to take place in late summer/autumn of the second year (due to school), some episodes occurred in the summer of that year, but no winter episodes were shown. It's entirely possible that season five will take place within a third year, but I'd have to see before coming to any conclusions.
Now, I apologize for the giant wall of text, and thank everyone who managed to read it. It's not a full chronology by any means (that would be impossible given the vagueness of the episodes), but it does give us a clear look into some of the major events and when they happened in relation to others. Please, if you see any errors or experience any confusion to let me know, and I'll do my best to clarify or fix whatever issues crop up.