In order to see some evidence of early Spanish expansion in North America, you just have to look at the many South-Western Cities of Santa Fe and San Francisco which are typical Spanish. These cities still have quarters where Spanish influence can still be seen.
In the 15th Century, Spain was the global leader in international trade, exploration and colonization.
One of the earliest evidence starts with Vasco Nunez de Balboa, who was the first known person at the time to discover and explore the wide expanse of the Pacific Ocean.
This was followed by Christopher Columbus, who despite being an Italian, was actually funded by the Spanish Throne in search of the Indies. He was able to discover the Bahamas and West Indies.
The Spanish eventually discovered modern-day Mexico and South America, where they were able to colonize large parts of the land.