While shoulder season weather conditions in your destination will help determine what kind of trip you’re taking (and can impact on-the-ground costs), the cost of airfare is typically connected to the season in your departure city—not the destination.
Specific shoulder seasons vary by region and country. Knowing when it’s spring or autumn in the hemisphere you’re planning to visit is a good start, but you’ll be much better off with details on the exact places you’re going.
Australia and New Zealand: mid-February–April, September–early December in Australia; March–May, September–November in New Zealand
Europe: March–April, October (in some parts of southern Europe where it stays warm later in the year, November may also be part of the fall shoulder season)
North America: March–May, September–November in the United States; May–mid-June, September–October in Canada; May, October–November in Mexico
South America: March–May (unless it’s Brazil, in which case “after Carnival” is more accurate), September–November
Caribbean: May–early June, late November–mid-December
Africa: Travel seasons vary significantly by region on this enormous continent. In East Africa, shoulder season is October–November. In North Africa it's October–November. In Southern Africa it's April and September–October. And in West Africa it's May and October
Asia: Travel seasons vary significantly throughout Asia, particularly between mainland countries and island nations. In Central Asia, shoulder season is July–August (great for mountain areas, very hot otherwise). In Japan it's March, late May–June, and September–mid-December. In China, it's mid-March–April ad September–October. In Taiwan, it's ate May–June, and in South Korea, it's April–May an dSeptember–October. In South Asia shoulder season is June–November. In Southeast Asia, it's March–May (hot but dry) and November.
West Asia/Middle East: March–May, September–November (except during religious festivals)