Where on Earth has the least changing temperature?
To answer the first question, yes, there are maps showing the seasonal temperature variation, for example here 

Unsurprisingly, the equatorial areas have the least variation because the angle of the sun relatively to the earth changes the least with seasons.

Regarding nice weather- I have worked as a geologist in northern South America and the Caribbean. In the lower lying areas (e.g. Aruba) I often thought it was too hot to be pleasant to work, but beachgoers think that was perfect weather. In the mountains it was often more pleasant, subjectively.
One city I visited where the temperature is remarkably stable is Medellin in Colombia - the "city of eternal spring". Situated close to the equator, temperature is always nice there (daily mean at 22 degrees throughout the year), but it can have a lot of rainy days.