I will only explain what Russia hoped to gain against the Ottomans and I am sure others will then explain what the Russians hoped for in other fronts, against Austro-Hungary for instance. The Ottoman factor was certainly influential in dragging Russia into WW1 and vice versa.
The Turkish Straits. When you look at the map of the era, the Straits are the only obstacle between Russia and the Mediterranean sea. An opening to the Mediterranean sea would yield both military and commercial benefits. Unsurprisingly, the British were initially very hesitant to grant the Russians their wish. In 1877-78 Russo-Turkish War, the British diplomatically intervened in favor of the Ottomans to ensure that the Russians do not take the Straits, as a permanent Russian base in the Mediterranean would threaten British interests in Cyprus, Levant, Egypt and even in India (short way via Suez Canal). In 1905, the Russians were bitterly beaten by the Japanese, who were allied with the UK. This, in a sense, showed the Russians that there was no point in forcing their way into Manchuria, China and the Far East in general. In 1907 they signed a treaty with the UK, settling their problems and turning their gaze towards potential gains elsewhere. By 1914 the British had other concerns like Germany and hence they seemed willing to sacrifice the Ottomans.
Eastern Anatolia. It is not a surprise that soon after the declaration of war the Russians were already in a position to launch an attack into Eastern Anatolia, then a part of the Ottoman Empire. If Eastern Anatolia were to be a part of the Russian Empire, this would strengthen the Russian position in Iran and protect the Caucasus against a potential pan-Islamist and pan-Turkish threat. Baku, now the capital of Azerbaijan, was a leading center of oil production so the area was certainly a place the Russians did not want the Ottomans (or anyone else for that matter) near.
Mustafa Aksakal's book The Ottoman Road to War in 1914 explains in detail how the alliance talks between Russia and the Ottoman Empire before the War failed. He argues that the Russians were not interested in a treaty with the Ottomans since this would clearly clash with their other objectives I mentioned. He also demonstrates how the Russian fleet knew in advance about the Ottoman attack on the Russian ports in October 1914. Instead of fighting the Ottoman fleet, the Russian navy simply let them bomb the ports thereby obtaining a casus belli to declare war on the Ottoman Empire.