Archduke Francis Ferdinand of Austria-Hungary did not commit suicide. He was murdered in 1914 in Sarajevo and the event started World War I. It was Crown Prince Rudolf who committed suicide with Baroness Maria Vetsera, not in Moedling, but in the imperial hunting lodge at Mayerling. Emperor Francis Joseph was so grieved that he ordered the lodge destroyed and instead built a small convent, which can be visited, on the site.
If you long for refreshments after sightseeing, drive to Kalksburg, where they serve the best millirahmstrudel (hot cheese strudel with raisins in cream sauce) and coffee mit schlag (whipped cream).