The largest effects of this war were that it ended the Spanish Empire and began the American one.
The Spanish Empire had been in a long, slow decline for more than two centuries and by 1898 had very few colonies under their control. By challenging and defeating the Spanish, the United States for the first time became a colonial power, taking Guam, the Philippines and Puerto Rico as US territories.
This ended easy Spanish access and influence in both the Caribbean and Asia and gave the Americans bases to project their military power (especially the US Navy) around the globe, securing resources and trade routes. The Spanish-American War ended the long era of Spanish naval power.
It also led to a long guerrilla war fought by Filipinos against their new colonial rulers, the Americans.