Now that Syrian forces have recovered Palmyra, where do they go from here? The campaign to retake Palmyra benefited from the inactivity on other fronts as a result of the ongoing cessation of hostilities (which excludes operations against the Islamic State and Jabhat al-Nusra), and certainly the Syrian government and Russia are claiming a significant victory. The Institute for the Study of War has an analysis of the Palmyra campaign as well as a useful map:
Syrian forces were already pushing toward Qaryatayn from the west; a column from Palmyra could provide additional pressure.
But Syrian reports also say that forces from Palmyra are advancing in the direction of Sukhna, which would suggest an effort to clear the road to Deir al-Zor. It is also claimed that the besieged forces in Deir al-Zor are pushing westward.
That will not be accomplished overnight. The distances are substantial, though mostly across open desert. The Palmyra campaign relied heavily on elite Syrian Army forces, Hizbullah allies, and Russian air cover. (As part of the Palmyra campaign, Syria has recovered the Tadmur Air Force base, giving Syrian and Russian aircraft a forward operating base.)
As an aside they have presumably also recovered the notorious Tadmur Prison, once the regime's most notorious.