The balkan politics of the great powers prior to WW1 is a complicated topic and difficult to explain in a short comment. First of, i would like to recommend a great book to you that is both easy to understand for beginners and very informative: The Sleepwalkers: How Europe Went to War in 1914 by Christopher Clark. It covers european politics pre-WW1 and especially the balkan politics.

So, to answer your question: What do you mean with "Self-Government"? Russia supported the independence of the balkan states because that weakened their enemy, the ottomans. Yet they didn't want them to be independent from russian influence. The russian support for balkan states always had the goal to strengthen russian influence in that region and to prevent the rise of any power that could block russias acces to the mediterranean sea. The idea of a common slavic background in all of these states was the ideological framework for the balkan politics, the russian government often tried to look like the protector of the suppresed southern slavic people to justify its sometimes aggressive foreign policy against the turks and the austro-hungarians.

The austro-hungarians had similar goals, trying to expand their sphere of influence to the south and especially to limit russian influence on the balkan states. They did not wan't to annex any of the balkan states that freed itself from the turks, that would leed to international conflicts the austrians could not win. I hope this very short overview helped you a bit, you better buy the book if you are really interested in the topic.

Basically what /u/Watermelonman1 said.

Austria (by this time Austria-Hungary) had naturally always kept its expansion eastwards, to expand the Austrian crown lands and to avoid conquering states of the Holy Roman Empire (Dissolved in 1806), which Austria came to be Holy Roman Emperor 99% of the time by the 1500's, with a few exceptions. The Holy Roman Empire states were commonly used as allies in Austria's wars and it was not worth alienating the rest of the Empire in pursuit of territory that could be gained elsewhere at lower cost.

Naturally the only place left for Austria to expand its influence was South towards the balkans, as a unified Germany was to the north, Russia to the East, and a Neutral Switzerland and a newly Unified Italy were to the west/southwest.

Russia also saw the Balkans as not only a perfect template to expand their sphere of influence, but also really their only means of possible expansion, as their natural rival, the Ottomans, while dubbed "The Sick Man of Europe" was being kept on life support by the British to prevent other powers from gaining points of interest that would threaten her trade.

The fact that the population was primarily Orthodox slavs allowed for the idea of a Russian led "Pan-Slavism" basically giving the Tsar a diplomatically valid reason to come to the "aid" of the numerous states against foreign agression. So while Russia technically supported self governing balkan states, the idea was that Russia would be the overarching power in the region, and these states could be used as allies in any future conflicts, or even become future acquisitions in the Russian Empire itself.

And well, the rest is history.