What are the evidences that the earth climate is getting warmer?
Besides rising surface air temperatures other indicators of global warming include -

ocean surface temperatures rising
ocean temperature at varying depths rising
global average sea level rising
Arctic sea ice extent decreasing
Arctic sea ice thickness and average age decreasing
Glacier retreat
Earlier Spring snow melt
Borehole temperatures rising
Earlier budding time of various plants
Less new record low temperatures
More new record high temperatures
Greenland losing ice mass
Antarctica losing ice mass
and etc

Every possible way we can observe and measure the changes we should expect with global warming shows results consistent with actual global warming.
Global Average Temperature as the go to single simple measure is a statistical construct, derived mostly from about 7,000 long running and up to 30,000 current weather stations recording daily minimum and maximums. It is a way to reduce complex change to a single simple number, for convenience. That metric may be a construct but it is constructed to show the extent of change to those recorded temperatures. In one sense it isn't a direct measurement of &quot;something&quot;, yet we can confidently expect this average to rise if the climate system is warming - and not rise if it is not.
These weather records allow other kinds of examination besides getting a global average temperature - mapping of regional changes, geographic ones and all the way down to the conditions at individual sites, including their suitability for inclusion in global temperature record calculations. Badly sited ones can be excluded, step temperature changes from changing location or changes to existing location can be adjusted for to achieve better accuracy.
These records don't exist in isolation - there is a broad understanding of climate and weather processes that goes with them. They are also supported by satellite data, such as microwave sounding that allow remote estimation of atmospheric temperatures; whilst they tend to show higher response to El Nino Southern Oscillation, the rates of warming are broadly consistent with weather station records.
It is a reliable observation that in between nearby weather stations sharing similar geography there are similar weather conditions, including similar maximum and minimum temperatures. Consistently different conditions in the spaces between would be noticed and we cannot expect such differences to be confined only in those between spaces; if there were significant variations in the temperatures and rates of change over time in the spaces between locations it would be happening at those locations as well and that would show up in the weather station records. What we find in those records is consistent with global warming.