Well, just about anything he wanted.
From the various canonical sources, we know very few things that connect Indy to the real world:
Indy traveled extensively in his youth, particularly in Egypt/North Africa, Europe, and Mexico. This was both with his family and during WWI.
He met a number of famous individuals, such as T.E. Lawrence and Bronislaw Malinowski
He enrolled in the University of Chicago in 1920, earning a a degree in linguistics.
In the classroom, we see him teaching on Neolithic England and Akkadian cuneiform and ceramic typologies.
Unfortunately, the only further knowledge we have of his education is that he was tutored by Dr. Ravenwood, an Arc of the Covenant enthusiast. At least that helps slightly narrow our scope. What we really want to know is:
"What would a UC student in the 1920s with an interest in linguistics and archaeology have studied?"
Let's start with what we know for sure.
Henry Jones, Jr. was certainly trained in North American anthropology. Indy's classes would have involved working at the Hopewell culture mounds in Liverpool, Illinois, under the direction of Fay-Cooper [Cole]. This would involve sick outfits and apparently some pipe-smoking. In the the late 1800s and early 1900s, a lot of university effort went into excavating the Hopewell and Mississippian mounds found throughout the Midwestern and South-Central US. As a linguist, Indy would have worked with Edward Sapir of Spair-Whorf hypothesis fame. Sapir's research at that time focused on native languages of the US and Canadian West.
Many of Indy's potential classmates went on to work in the US. Fred Eggan would become president of the American Anthropological Association after becoming known for his application of social anthropology to native American tribes. Physical anthropologist George Neumann went on to establish the University of Illinois' first archaeological collection. One can imagine the young Henry getting cozy with "one of the most popular young women in society" Mary "Betty" Frey before she got hitched and seemingly left the archaeology business. Fellow student Paul S. Martin, hanging with Drs. Sapir and Cole, would contribute enormous amounts of research on the American Southwest to the Field Museum.
Two possible classmate took a more Indiana Jones path. Richard G. Morgan, seen excavating with Ms. Frey above, was influential in our understanding of the Hopewell culture, but was fired from his position as a suspected Communist. Indy would have the same thing happen 9 years later. Charlotte Chapman Gower, seen with a brush in the second linked pic, was captured by the Japanese during WWII. She made it back safely to become a Marine and CIA agent.
But we know that's not all Indy was into, and that's just fine. The Oriental Institute at UC was founded by James Henry Breasted one year before Indiana showed up. Indy would have hopped right on with Breasted's expeditions to Egypt in the following years, contributing to the epigraphic survey of Medinet Habu. Breasted focused on Egypt, but also could have taught Indy Hebrew, Akkadian, and general history of what would have been called at that time the beginnings of Western Civ in Mesopotamia and its neighbors. William Gardner Hale and Paul Shorey would have supplimented this with their then-revolutionary studies of syntax and language strucutre. At the same time, Frederick Starr was influential in the Anthropology department; his research included Japan, Mexico, central Africa, and SE Asia. The already mentioned Ms. Gower studied Sicilian society, and the department at that time was following Starr into Mexico. Dr. Fay-Cooper Cole, though he did work in the US, also published on Polynesian societies. And of course there's the history and literature departments where one could study Europe's past. Really the only gap would have been South American studies. It would certainly have been possible, but it wasn't not a focus of the period.
TL;DR Indiana Jones would have definitely studied North American archaeology, Egyptology, and linguistics. His hypothetical teachers and classmates were the top current and future scholars. The UC anthropologists were studying cultures all over the world. Also I find some cool bits about other 1920s UC students who were almost Indiana Jones.
