Why did Russia support Serbia in the years leading to World War I?

This, as is history's wont, a question with quite a few events leading up to it.

Pan-Slavism - the ethnic element

As mentioned in the comments on the OP, Pan-Nationalism was all the rage at the time. In response to the Frankfurt Assembly and the Unification of Germany some Slavs felt that their rights were in danger as a result: https://en.wikipedia.org/wiki/Prague_Slavic_Congress,_1848

Pan-Slavism never really took off for the Czechs, but in the Balkans were the Slavs had been dominated by the Ottomans and Austrians for centuries, the idea had great appeal and gained traction. The Balkan Slavs naturally looked to Imperial Russia, as the great "Slavic" power, as a natural ally; in turn the Russians saw them as allies against its two rivals, the Ottoman Empire and Austria.

Orthodox Christianity Versus Islam - the religious element

There was also the religious element. The Balkans had large Christian/Orthodox and chafed under their Muslim overlords. Russia, an Orthodox nation thus had strong cultural ties to the Balkans and many Orthodox Christians lived under Ottoman rule and were treated as second-class citizens.

The Russians contended that the Treaty of Küçük Kaynarca granted them the right to protect Orthodox Christian in Turkey. This contention was one of the tensions that led to the Crimean War.

Article 9 of the 1856 Paris Peace Treaty, concluded at the end of the Crimean War, obliged the Ottoman Empire to grant Christians equal rights with Muslims. Before the treaty was signed, the Ottoman government issued an edict, the Edict of Gülhane, which proclaimed the principle of the equality of Muslims and non-Muslims,[7] and produced some specific reforms to this end. For example, the jizya tax was abolished and non-Muslims were allowed to join the army

However, Ottoman reforms were incomplete. In areas with Christian majorities, authorities often supported abuse as a way to keep Christians in check. This led to a number of crises such as the 1860 Druze-Maronite conflict, & the Cretan Revolt.

Geopolitical stresses - the great power element

There were geopolitical motivations as well. The Russians and the Ottomans had been in conflict from the 16th right up to the 20th Century. In fact, this rivalry is still evident today, although in past months tensions have cooled. Russia and Turkey shared a long border in the Caucuses and this caused some friction. The Russian-Ottoman conflicts were one of the longest running series of conflicts in Europe.

After 300 years of living under threat from Ottoman domination, the Eastern Question was opened after the Treaty of Küçük Kaynarca marked the retreat of the Ottomans, and granting Russia access to the Black Sea.

The Eastern question was how would the balance of power established at the Treaty of Vienna in 1815 be maintained. Treaty of Vienna. Russian wanted a free hand to expand into Southern Europe, as well as access to warm water ports in the Black Sea, and ostensibly, access to the Mediterrenean. The British were determined to forestall this, which formed part of the 'drift' to the Crimean War.

The Concert of Europe began to completely come apart after the Unified Germany defeated Austria, then France in the Franco-Prussian War of 1870 (an event that would also play no small part in shaping events leading up to the First World War). In order to keep France contained on the continent, Bismarck created the Three Emperor's League. To counter this, France fomented uprisings and supported indepdence movements, including in the Balkans - Russian countered this by using the emerging Pan-Slavic ideal - that all Slavs should be united under Russian leadership.

The Stew comes to a boil

From 1804 - 1817 Serbia revolted at Ottoman rule for a number of factors: The Serbian Revolution

Encouraged by the Russian Empire, the demands for self-government within Ottoman Empire in 1804 evolved into a war for independence by 1807.

Another role model was the Russian Empire, the only independent Slavic and Orthodox country, which had recently reformed itself and was now a serious menace to the Turks. The Russian experience implied hope for Serbia

The Serbian Revolution ultimately became a symbol of the nation-building process in the Balkans, provoking peasant unrests among the Christians in both Greece and Bulgaria.

In 1875-76, as a result of deteriorating Ottoman administration in the Balkans a Bosnia, Bulgaria and Herzegovina revolted against the Ottoman Empire. Serbia also declared war on the Ottomans on the 30th of June in 1876. Russia, justifying its position on the grounds of "fraternal allegiance" under Pan-Slavism, also declared war, beginning the Russo-Turkish War 1877-1878. Thus the Great Eastern Crisis was under way.

The decision to increase taxes for paying the Ottoman Empire's debts to foreign creditors sparked an outrage in the Balkan provinces, which culminated in the Great Eastern Crisis (1875–78) and ultimately the Russo-Turkish War (1877–78) that provided independence or autonomy for the Christian nations in the empire's Balkan territories, with the subsequent Treaty of Berlin in 1878.

Russian designs on the Bosphorus, The Dardanelles and access to the Mediterranean

Since Peter the Great, Imperial Russia long sought to secure rights of Navigation from the Black Sea into the Mediterranean. Russia wanted full rights not just for commerce but for its warships - something that still motivates Russian foreign policy today, as evidenced by Russian support of al-Assad in Syria to maintain control of its port at Tartus.

Russia was motivated to find reliable client states in the Balkans to hedge against its rivals, Austria and Turkey. Whilst it did hope for complete possession of the Straits, or to at least secure passage for Russian ships whilst denying it to other powers, no other great powers would concede such a prize to Russia. The British acted consistently to oppose Russian influence in the Eastern Mediterranean.

Conclusion

Russian interests in the Balkans seem to have been primarily:

• Retention and expansion of right of navigations through the Dardanelles;

• Territorial expansion at the expense of the declining Ottoman Empire.

• Creating reliable client states to contend with its two major regional rivals: Austria and Turkey, under the guise of pan-Slavism, or protecting Orthodox Christians.

  The Russians would have been keen to avoid a repeat of the disastrous defeat of the Crimean War that saw them lose the right to host warships in the Black Sea.

The Russians also viewed the Balkan states as a buffer against Austrian influence. This is why they supported Serbia in the weeks leading up to the outbreak of World War One.

The Tsars had also long held designs on Constantinople and the Bosphorus, which would further serve as a bulwark against incursions from powers like Britain, whom was its greatest geopolitical rival.

The Russians also slowly dismembered Ottoman control of South-Eastern Europe as part of their aims to reducing the Ottoman Empire to that of a petty vassal state.

Thus we can conclude that Russian Balkans policy was driven by a goal of territorial expansionism and imperial ambition. This is a policy that continues to this day.