Recommended fasts include Mondays/Thursdays, the three white days each month, six days of Shawwal, and Arafah/Ashura days — avoid fasting on Eid and Tashreeq days.
Voluntary fasting after Haji is an excellent way to continue building on the spiritual foundation established during the ibadah haji. The most significant post-Haji fasting opportunity is the six days of Shawwal — Nabi (shallallahu alaihi wa sallam) said: 'Whoever fasts Ramadhan then follows it with six days of Shawwal, it is as if they fasted the entire year' (Muslim). Since Haji occurs in Dzulhijjah (after Shawwal), this applies if you fast the following Shawwal.
Year-round sunnah fasts include: Mondays and Thursdays (Nabi's regular practice, as these are days when deeds are presented to Allah), the three white days of each lunar month (13th, 14th, and 15th — Ayyam al-Beed), the Day of Arafah (9th Dzulhijjah, for non-jamaah haji), and the Day of Ashura (10th Muharram, with the 9th or 11th). Important: fasting is prohibited on the two Eid days and the three days of Tashreeq (11th-13th Dzulhijjah), so new jamaah haji should not attempt to fast immediately after their Haji rituals conclude.
Source: Sahih Muslim; Sunan al-Tirmidhi; Imam al-Nawawi, Riyad al-Salihin