Reconquest is the easy part
The Balkan states as of 1899
This map from a turn-of-the-century British atlas shows what everyone was getting ready to fight over. Turkey-in-Europe is the large area colored green, covering modern Albania, Kosovo, Macedonia, and Thrace (which extends along the Aegean coast from a point north of the island of Thasos eastward to Constantinople itself, at the entrance to the Black Sea). The map adds Eastern Rumelia to the territory of Bulgaria, although this wasn’t official until 1908 (the same year that Austria-Hungary annexed Bosnia, which is why it's a different color here).

Also, note that Serbia is spelled Servia, although the only language in which it’s pronounced that way is Greek. The British seem to have adopted the Greek pronunciation early in the 19th century—perhaps under the spell of Philhellenism—and to have taken quite a long time to get over it. (Ditto for the Americans, who no doubt followed the British lead.) World War I seems to have brought Serbia to everyone's notice at last, and the v-spelling finally disappeared from maps and texts written in English.

The Balkan League intended to push the Ottoman empire all the way out of Europe. If you compare the sizes of the Balkan countries on this map and on the postwar maps coming up later, you’ll see how much larger each one became, although the Turks would hold onto a small patch that included their capital, and would still control both sides of the passage between the Aegean and Black Seas.

One factor that hastened the drive to complete the Balkan League pact was the knowledge that the Ottoman empire’s military forces were involved at the time in a war with Italy over Libya. That had begun in 1911 and was still dragging on as the Balkan countries put their league together in 1912. Other Ottoman troops were engaged in putting down an uprising in Yemen. It was desirable to get the war started before the Turks could pay full attention to defending their European possessions.

The war began in in the fall of 1912 when Montenegro declared war on October 8. Bulgaria and Serbia followed on the 17th, and Greece on the 18th, first proclaiming its annexation of Crete (and finally admitting the duly elected Cretan representatives to sit in the Greek parliament). The Italo-Turkish War, in the meantime, had ended only on October 15.
Battle of Sarandáporo (artist's conception)
Crown Prince Constantine commanded the main Greek army, which marched across the border into Macedonia; a smaller force headed for Northern Epirus (or as it was called in certain other places, Southern Albania). The prince and Venizélos came in conflict early in the campaign. On October 22, the Greek army captured the strongly fortified mountain pass of Sarandáporo, not far west of Mount Olympus, which was the main border defense in their path. (You shouldn't regard this patriotic lithograph as an accurate rendition of the battle, but remember that Greeks hungered for good military news—this was the army's first battle since its 1897 humiliation by the Turks.) Beaten fair and square this time, the Ottoman forces retreated northward, and Constantine intended to pursue them toward the Macedonian town of Monastir. That was the militarily appropriate procedure: engage the enemy’s force and defeat it, no matter where it happens to be, before attempting to occupy any of their territory. But Venizélos was more concerned with the political necessity of taking and holding the Macedonian territory that Greece intended to claim, particularly Macedonia’s biggest city and only seaport: Salonika.

To secure Salonika, the army would need to move northeast rather than stay on the trail of the retreating Turks. If the purpose had been to keep the city away from them, the prince would probably have been right. But as Venizélos was well aware, Salonika really had to be defended against Greece’s ally Bulgaria. Secret messages from the Greek embassy in Sofia had informed him that a Bulgarian army was already heading for the city. The crown prince, as army commander-in-chief, was legally under the authority of the defense ministry, and therefore of the prime minister—but he was, of course, also the son of the monarch, and the future monarch himself (a destiny that he was to realize much sooner than anyone expected). There was a good deal of ill-natured arm-wrestling in the interaction between him and Venizélos, and it was resolved only when King George ordered his son to follow the prime minister’s direction and pursue the political agenda.
Crown Prince Constantine entering Salonika
The Greek army entered Salonika on November 12, barely a day ahead of the Bulgarians. When they arrived the next morning, however, Constantine agreed to let a token Bulgarian unit enter the city. It was soon enlarged (contrary to promises given) to 15,000—there were 25,000 Greek troops in the city—and began to act as if the two countries were coequal occupiers. Venizélos blamed the prince for this awkward situation, and the whole episode (starting from the quarrel over which way the army should go) is considered the beginning of the antagonism between the two men. It became bitter and permanent, and in time inflicted on the Greek body politic a split—called the National Schism [Ethnikós Dichasmós]— that was not to heal for generations.
The Avérof in the Battle of Élli
The Greek navy succeeded fully in its assignment to block any transfer of troops from Asia Minor or the farther reaches of the Ottoman empire, which had few railways at the time, making transportation by sea the only possible way to reinforce their beleaguered armies in Europe. Admiral Koundouriótis’ fleet maintained complete control of the eastern Mediterranean; naval expeditions liberated one Greek-speaking island after another from Turkish rule while with the main part of the fleet Koundouriotis kept the Ottoman navy bottled up in the Dardanelles, and defeated it soundly on the two occasions (the battles of Élli, December 16, 1912, and Lémnos, January 5, 1913) when it tried vainly to break out. His most potent weapon was the armored cruiser Avérof, which was faster and better armored than any vessel the Turks had. The Greek sailors may have been better trained as well: Wikipedia reports that in the battle of Lémnos, “the Ottoman ships achieved an excellent rate of fire, firing about 800 shells, but with dismal accuracy.” Only two shots hit the Avérof, and neither of the two Greek battleships in the fight was hit at all.

As noted elsewhere on this website, the Avérof has an interesting history.
Serbian army entering Skopje (October, 1912)
Serbia scored big successes on the battlefield, most notably at Kumanovo, where the Ottoman commander, believing that the Serbs were unprepared, rashly launched a preemptive attack against superior numbers. The Turks lost heavily, and had to retreat south, yielding most of northern Macedonia including Skopje, Macedonia's second-largest city and modern capital. A Serbian army then moved westward, taking much of Albania without opposition, and captured the city of Durrës (Durazzo) on the Adriatic coast.
Bulgarian postcard of Lule Burgas victory
Bulgaria sent some of its troops south into Macedonia (where, as we’ve seen, they narrowly lost the race for Salonika to the Greeks), but concentrated its efforts in Thrace, to the southeast. They were assisted by faulty Ottoman intelligence, which expected a Greek amphibious attack on the Dardanelles and deployed part of their available forces to repel it. The attack never came, and in the meantime the Bulgarians defeated the Ottoman forces at Kirk Kilisse and, in a fierce battle that lasted five days, at Lule Burgas.
Ottoman troops in retreat from Lule Burgas
By December of 1912, the Ottomans had already lost virtually all of Turkey-in-Europe except Istanbul, protected by a line of trenches at Chataldja, less than 20 miles out of town. The Bulgarians came that close to the city, but couldn’t get past this final defense, and were repelled with heavy losses when they tried.

The only other parts of Europe still in Turkish hands were three cities under siege: Scutari (Albanian Shkodër) in northern Albania, besieged by the Montenegrins with some help from Serbia; Yanina (Greek Ioánnina) in Northern Epirus (or Southern Albania, depending on whose point of view one adopts), besieged by the Greeks; and Adrianople (Turkish Edirne) in Thrace, besieged by the Bulgarians. (The alternative names I’ve given for the three cities indicate who owns them today.)
The powers try to keep the lid on
As usual, the powers of Europe began to feel anxiety that events in the Balkans were getting out of hand—in other words, threatening to conflict with their own desires. Austria-Hungary, worried lest Serbia, perhaps as a Russian partner, gain an outlet to the Adriatic, which was Austria-Hungary’s only access to the oceans of the world, was suddenly overflowing with concern for the national interests of the Albanian people. So was Italy, which didn’t want any additional rivals in the Adriatic. The British and French, seeing how quickly and easily the Greek navy was liberating Aegean islands, realized that control of the islands that defended the Dardanelles might wind up in Greek hands instead of their own. Both Greece and Bulgaria had designs on Constantinople. So did Russia.

The Bulgarians and Ottomans signed an armistice (proposed by the latter) on December 3, and Serbia and Montenegro also agreed to it, though Greece declined to sign, and carried on its siege of Yanina throughout the brief peace that followed. But negotiations for a more permanent peace took place in London.