This is a pretty tricky question. I'm not 100% convinced the 1953 discovery of the structure of DNA changed ideas of heredity all that much. The beginning of the 20th century saw the 're-discovery' of Mendel's pea experiments which really boosted the field of genetics. So, by the 1950s scientists had a pretty good idea of how simple genetic inheritance worked, although the physical structure of the genetic material was still a mystery. There was also tension between ideas of evolution; extremely simplified this was Darwinists vs. Lamarckians.
So basically they kind of had the idea that there was genetic "stuff" like chromosomes etc, but really until 1953 there was no molecular genetics in the same way there is today.
If you are interested in the history of genetics, James Watsons book "The Double Helix" is a fascinating examination of science at the coal-face. It shows exactly how much the race towards this discovery was very much shooting in the dark.
There are many great primary sources about the tension between Lamarckian's and Darwinists in America during the first half of the 20th century. Look for stuff by anthropologists like Charles Davenport at Cold Spring Harbour and compare it to the much more liberal Franz Boas.
As far as information on history of genetics in particular most of my facts come from "In Pursuit of the Gene: From Darwin to DNA" which is by James Schwartz. Key players to look at prior to Crick and Watson could be Darwin, T.H. Morgan and De Vries. Mendel was interesting and crucial, but not really until after he was dead. He was a little too obscure to be influential.
By the 1910s-1920s there was already strong agreement amongst biologists that heredity worked more or less the way we know it to today. That is, they knew it was caused through the meiosis of gonadal cells which allowed the transference of chromosomal material (i.e. DNA) and accorded with Mendelian "laws" of heredity. The work is took to get to that point is a long story from the mid-19th century onwards, but the person who probably has the largest role in consolidating this opinion, and establishing that the physical basis of Mendelian heredity involved the chromosomes, was the American biologist Thomas Hunt Morgan. Morgan and his students' work on fruit flies (Drosophila melanogaster) helped bridge the gap between various heredity theories (such as Mendelism) and observable cellular biology.
The discovery of the structure of DNA did not change our understanding dramatically, at least not immediately. It did later allow for more detailed probing of heredity on the level of individual genes, but even that was a long way off. If anything, what the structure of DNA really elucidates is not so much how heredity works (because again, that is covered quite well with chromosomal theory and the theory of the gene, which well predated the discovery of the double helix structure), but how the cell works in general (the way in which DNA serves as the regulator of cellular function and activity). That DNA was the "hereditary molecule" was actually realized prior to the discovery of its structure (and is what motivated the work to discover its structure).
My favorite short book on the history of heredity is Peter Bowler's The Mendelian Revolution, which covers hereditary thinking from the 17th century or so until the "rediscovery" of Mendel's work in the early 1900s. Note that much of this is not distinct from evolutionary thinking for much of its history.
Separately if you are interested in Morgan's work, which is fascinating, the best book is Robert E. Kohler's Lords of the Fly (groan-worthy title but there you have it). It focuses on the operation of Morgan's laboratory, specifically on the choosing and exploitation of the fruit fly as a "model organism" for genetics research (its life-cycle syncs very well with academic calendars, as one of its many benefits).