Remarks in italics are not taken from explicitly-stated events in the canon material. They are my own speculations, logical inferences, gap-fillers, and extrapolations based on fragmentary references and passing mentions in the original sources.
June 1925 — The newly-minted “Professor Jones” gets his first job after graduating — teaching a summer course on Britain’s megalithic monuments at the University of London.
He discovers he has a brilliant but difficult student, Deirdre Campbell, who is the daughter of his department head, Dr. Joanna Campbell.
Mid-August, 1925 — At the conclusion of his summer course, Indy is invited by Dr. Campbell to join her and Deirdre on an excavation of St. Ninian’s Cave on the southwest coast of Scotland to look for a golden scroll that proves the legend of Merlin the wizard had basis in historical fact. After a series of violent run-ins with unknown assailants, Indy, with the help of Deirdre and Jack Shannon, gradually uncovers a plot by a modern day druidic cult, the Hyperboreans. The leader of the Hyperboreans, Albert Powell (wealthy member of Parliament and, it turns out, Deirdre’s half-brother) plans on using the scroll in a ceremony to unleash the power of the Omphalos, which he recently arranged to have stolen from the museum in Chicago. The scroll is not at St. Ninian’s Cave, but they find a clue as to where it actually is — in an old convent near Stonehenge. Powell heads out to recover it, leaving Indy, Shannon, and the Campbells to die in the cave he has filled with toxic gas. Three of them escape, but Joanna Campbell succumbs. They actually find the scroll in the convent ahead of Powell, but are captured by him not long after, and discover that the midnight ceremony Powell is planning includes a human sacrifice in the form of Deirdre. It turns out the Omphalos that Powell stole is a replica, and the ceremony fails. Powell perishes by falling into a massive bonfire.
Under the pressure of these few weeks, Indy and Deirdre find themselves in a passionate romance.
In Scotland, Indy is presented by the townspeople with a Webley .455 revolver. In freeing Deirdre from the cult ceremony, Indy makes his first peacetime self-defense kill with the Webley, and it “sickens” him, but from this point on, he rarely ventures out on his expeditions without a handgun.
Fall 1925 — Indy begins teaching full-time at University of London, mostly courses in Celtic archaeology. He admits it’s not his favorite area, and he struggles with the Celtic language despite his facility with other languages. (Perhaps it’s all too closely associated with his father’s Holy Grail obsession.) He spends his off hours rambling around the English countryside, for which he develops a deep affection. It is around this time, in the New Forest, that Indy meets Gale Parker, an academic and adventurer, basically a female version of himself (except she can fly a plane.)
January 1926 — Indy finds out he is not scheduled to teach any classes for the upcoming semester, but will instead assist on a major University of London expedition to Tikal, Guatemala.
February 21, 1926 — Under the leadership of the new head of the archaeology department, Dr. Victor Bernard, the expedition begins moving from the coast to the Guatemalan interior. Deirdre Campbell, whose relationship with Indy has cooled in the aftermath of the previous summer, is also on the expedition.
March 7, 1926 — After some exciting initial discoveries in a Mayan pyramid, the expedition is disrupted by graverobbers, and they are lucky to escape alive.
March 8-14, 1926 — Indy receives treatment for a gunshot wound suffered in the robbery attempt (it wouldn’t be his last), and the failed expedition makes its way back to New York via train.
Mid-March, 1926 — Indy, arm still in a sling, attends the opening night of Marcus Brody’s highly controversial National Museum exhibition displaying evidence that ancient Old World civilizations had been to the Americas long before Columbus. Brody shows Indy several pages from a journal written by his old friend, noted explorer Jack Fawcett, who has gone missing in the midst of searching for a lost city, Ceiba (popularly known as “El Dorado”), deep in the Amazon jungle. Fawcett believed it was inhabited by fair-skinned people of Celtic origin. Indy and Deirdre decide to go in search of Fawcett.
Late March, 1926 — On the eight-day ocean voyage that takes them to Brazil, Indy and Deirdre rekindle their romance. As the ship arrives in the bay at Rio de Janeiro, Indy and Deirdre are married by the ship’s captain.
Various clues and encounters with practitioners of the mysterious Candomble religion indicate that Fawcett is being held captive as “breeding stock” in the lost city of Ceiba. The inhabitants of the city have the ability to “veil” — to become like shadows, effectively invisible. High-ranking members of their society (which does appear to have Celtic origins) also have the ability to veil the city itself from outsiders.
Early April, 1926 — After days of traveling by seaplane and on foot deep into the Amazon jungle, Indy and Deidre are taken captive by Ceiba’s inhabitants. The veiling causes everyone in Ceiba to exist in a semi-dream state. While their fate is being decided in shared dream with the Ceiban High Council, their physical bodies are busy escaping, plunging off a cliff into the river below. Pursued by the Ceibans, Indy, Deidre, and Fawcett make it back to the lake with the plane. After taking off, they discover the fuel tanks have been drained by a hostile tribe, the Morcegos — the plane plunges into the jungle, killing Fawcett and Deirdre. Indy is rescued from the jungle after several days, suffering from multiple injuries and typhus. He has no memory of his time in Ceiba or the crash that killed his wife of only a few weeks.
Summer 1926 — Indy is still grieving for Deirdre. At some point during time away from his university duties, he examines the work of the ancient American Mound Builders along the Missouri River.
He also pays an extended visit to his old mentor, Abner Ravenwood in Chicago, perhaps assisting Abner in his work. During one of their discussions, Abner cautions Indy that in the overlapping areas of mythology, occult, and archaeology that they both specialize in, it is important to remain skeptical and rational in front others, no matter what one may witness with his own eyes. Under the circumstances, Indy is grateful to have to distraction of working with Abner, and basks in the hero worship Abner’s teenage daughter, Marion, lavishes on him. In a complete lapse of judgement, Indy seduces Marion. Abner finds out, and this leads to a permanent break in their friendship.
January 1927 — Indy travels to a conference in Dublin to present a paper on Celtic influences in New England.
May 1927 — Depressed over recent events, and still feeling that Celtic archaeology is a bad fit for him, Indy resigns from the University of London and returns to Chicago to investigate possible employment at his alma mater. He finds his old friend Jack Shannon running a jazz club, and reluctantly involved in the organized crime that comes with the territory.
June 1927 — Jack Shannon, due to the pressures of his new lifestyle, has found religion and invites Indy to his church to hear a lecturer, Dr. Zobolotsky, who claims to have discovered Noah’s Ark, and is still in possession of a small piece of it. After the job possibility at the University of Chicago falls through, Indy finds himself agreeing to join Zobolotsky and his daughter, Katrina on an expedition to Mount Ararat in Turkey to document the existence of the Ark. After Shannon’s brothers are murdered in a mob hit, he decides to go on the expedition as well.
June 21, 1927 — Indy and Shannon leave Chicago by train to join the Zobolotsky expedition. They are trailed the whole time by Bolshevik agents, who feel that proving the Ark’s existence would undermine the anti-religion Bolshevik government.
Early July, 1927 — The expedition arrives in Istanbul and heads for and heads for Mount Ararat.
Mid-July, 1927 — Partway up the mountain, the two tailing Bolsheviks shoot and kill Zobolotsky, and are themselves killed by the local Kurds. Indy, Shannon, and Katrina continue up the mountain, and finds the Ark — and a bunch of Janissaries, a fanatical Turkish religious group who believe finding the Ark will trigger the apocalypse. Gunfire ensues, causing an avalanche which buries the Janissaries and the Ark, probably forever. 
August 1927 — Shannon and Katrina become engaged and head off to California. Indy is hired by a small, unnamed college in New England.