I'm reading The Sleepwalkers: How Europe Went to War in 1914 by Christopher Clark. Either I missed something or it was omitted, but I don't understand how Russia and Austria-Hungary's foreign policies and desires conflicted with respect to the Balkan region. For example, Clark states that the Franco-Russian Alliance, formed in the 1890's, partly stemmed from Russia's desire to block Austria-Hungary, who was part of the Triple Alliance, in the Balkans. Can anyone provide specifics?
This was an era of peak nationalism, so it probably helps if you think ethnically.
Let's compare a political map of Europe in 1900 to an ethnic one, taking special note of eastern and southeastern Europe:
Source Friedhelm_der_VI on r/Maps
The Czars in Russia saw themselves as the heads of the most powerful Slavic state in the world. As such, they felt themselves honor bound to stick up for other Slavic peoples. If they were in neighboring states, perhaps they'd be treated better if that land were part of the Russian Empire. If they were part of someone else's state and didn't want to be, perhaps Mother Russia should help them leave. If they had their own state already (as the time of the map above, only Bulgaria and Serbia), an alliance with Russia would help keep them safe and independent.
Meanwhile the Austro-Hungarians were heirs to an empire that was built in a bygone era where European rulers didn't really care much about the cultures of the lands they were conquering. The ruling family was German, but only about a quarter of their citizens spoke German. Perhaps another fifth spoke Hungarian (not even an Indo-European Language), while about half of the population spoke one of a myriad of different Slavic languages. There were also a few Romanians and Italians (2 very different Romance languages).
In a new era where people politically identified ethnically ("Nationalism"), this kind of state was inherently unstable, and took a lot of political (and no small amount of miliary) effort to hold together. Eventually the Hungarians had to be let into political power as partners (hence "Austro-Hungarian), and the heir to the Imperial throne was making noises about cutting the Slavs in on the deal too.
So it should be pretty obvious that where Slavic Nationalism was concerned, these two Empires had diametrically opposing interests. So as long as that was the animating political issue in the Balkans, they were going to be on opposite sides.
Of course, they could always find common ground on how to deal with Slavic Nationalists within the Ottoman Empire, as long as the sticky issue of who gets the liberated Slavic territory can be avoided.
At a fundamental level, both Russia and Austria-Hungary were imperialist, multi-ethnic states too close to each other. Any specific events, like the assassination of Archduke Franz Ferdinand, have to be seen under the lens of this rivalry.
A century before the time you asked about, Austria stood with one leg in the remnants of the Holy Roman Empire, and with the other leg in the Balkans. Prussia and Austria fought for control of Germany and Prussia won. A few decades later Germany and Austria became allied, yet the way to dominate the smaller German states was blocked.
Czarist Russia had been on a steady growth from Muscovy towards the east, south, west, and north. Their expansion into Siberia is well known, yet they also went into Finland, Poland, and Ukraine, to name a just few. Russia saw itself as the natural leader and protector of the Slavic peoples in the area, but it was also ready to conquer non-Slavic peoples. Austria-Hungary did include some Slavic peoples.
The causes of the First World War are quite complicated, and defy a simple explanation like "nationalism," which in itself is not a sole cause nor sole symptom to any war. It's best to refer back to a contemporary source.
Liddell Hart put it best:
The fundamental causes of the conflict can be epitomized in three words - fear, hunger pride.
Peeling the onion back a layer, he has this summary:
Fifty years were spent in the process of making Europe explosive. Five days were enough to detonate it. [..] On one side we should have to trace the influence of Prussia on the creation of the Reich, the political conceptions of Bismarck, the philosophical tendencies in Germany, and the economic situation - a medley of factors which transmuted Germany's natural desire for commercial outlets, unhappily difficult to obtain, into a vision of world power. We should have to analyze that heterogeneous relic of the Middle Ages known as Austria-Hungary, appreciate her complex racial problems, the artificiality of her governing institutions, the superficial ambitions which overlay a haunting fear of internal disruption and frantically sought to postpone the inevitable end.
On the other side we should have to examine the strange mixture of ambition and idealism which swayed Russia's policy, and the fear it generated beyond her frontiers, especially among her German neighbors, perhaps the deadliest of all the ingredients in the final detonation. We should have to understand the constant alarms of fresh aggression which France had suffered since 1870, study the regrowth of confidence which fortified her to resist further threats, and bear in mind the wounds left in her side by Germany's surgical excision of Alsace-Lorraine. Finally, we should have to trace Britain's gradual movement from a policy of isolation into membership of the European system and her slow awakening to the reality of German feeling towards her.
To look specifically at Russia and Austria-Hungary, you have to look at the Balkans. Turkey was weak and being forced out of Europe. Bulgaria, Serbia and Greece went after Turkey to peel off Macedonia. Austria's "meter of concern" rose depending on how much Serbia benefited from the spoils, because the inter-play between Serbia and Austria-Hungary's ethnic situation was complicated. Serbia could become a rallying cry for the non-trivial slavic elements within the Empire, and Vienna felt that settling the issue in a foreign war with Serbia was better than an internal revolt. And so Austria-Hungary began to eye a war as a way of opening a social safety valve.
Russia was also thinking in these terms. Internal unrest had been quite serious in the past years and was building up again. The "golden ring" would be its dream of capturing the Dardanelles from Turkey. Her Balkan clients had faltered in delivering that dream (or at least showed they could never deliver such a lofty prize), and by this point Germany was building up Turkey's army, and so many Russian ministers were inclined to believe that a general European war would knock back Turkey (and Germany) enough to achieve that objective, while also staving off an internal revolt.
Indeed most of Europe was thinking in terms of war by 1914. The pro-war parties in Germany were becoming intense (despite causing the pro-peace parties to coalesce around the Socialists). Poincare in France declared that France would not seek war, but was not afraid of it either. Hart describes a sense of fatalism descending upon the continent at that time. He describes the situation as gunpowder having been strewn everywhere, with many inclined to light a match to meet their objectives.
Liddell Hart, B.H. A History of the First World War. Pan Books, 1970.