Do I pray sunnah prayers while traveling?

The regular sunnahs are dropped

At home, a Muslim usually prays extra rak'at around the five prayers (the rawatib), such as two before Fajr, four or two before Dhuhr and two after it, two after Maghrib and two after Isha. On a journey the Prophet ﷺ normally did not pray these. Abdullah ibn Umar was once asked about people praying sunnah after a shortened prayer on a trip. He said that if he were going to pray extra, he would rather have prayed the obligatory prayer in full, and that he had travelled with the Prophet ﷺ and seen him keep to two rak'at (Sahih Muslim 689).

So skipping them is not laziness. It follows the sunnah of travel, which is about ease.

Keep the two rak'at before Fajr

Aisha said the Prophet ﷺ was never more careful about any voluntary prayer than the two rak'at before Fajr (Sahih al-Bukhari 1163). He also said these two rak'at are better than the world and everything in it (Sahih Muslim 725). Scholars agree he kept them while travelling too, so try not to drop them, even on an early flight.

Keep Witr as well

Witr is the odd-numbered prayer at the end of the night, after Isha. The Prophet ﷺ prayed it on journeys as well, sometimes on his riding animal. The Hanafi school even treats Witr as wajib (required). A single rak'at of Witr before you sleep in a hotel is enough.

Other voluntary prayers are fine

Night prayer (tahajjud), Duha, the two rak'at when entering a mosque, and any general extra prayer are all allowed while travelling. The Prophet ﷺ prayed voluntary prayers on his mount facing whichever way it was going (Sahih al-Bukhari 1099), which is why many scholars allow sitting voluntary prayer on a train or plane:

For the obligatory prayers on a flight, see Can I pray on a plane? For shortening itself, see What is qasr prayer?