If I had a hot cup of tea and I added to it cold milk, what would happen is that tea will lose some of its thermal energy to the milk. But why do we notice that our liquid is no longer as hot as before adding milk? After all, the total thermal energy of the system (the cup) is conserved even if there has been a transfer of heat between the two. Suppose the cup is an isolated system.
Before mixing the average kinetic energy of the molecules which make up the tea is greater than the average kinetic energy of the molecules which make up the milk.
This is restatement of the fact that the tea is hotter than the milk as the temperature of a substance depends on the average kinetic energy of the molecules which make up the substance.
When you mix the tea and the milk together the molecules of the tea and the milk collide and eventually reach an average kinetic energy (temperature) which is less than the average kinetic energy (temperature) of the tea before mixing and is greater than the average kinetic energy (temperature) of the milk before mixing.
So the collisions between the molecules redistribute the kinetic energy amongst the molecules and what is called thermal equilibrium is reached.
If there is no loss of heat or chemical reaction then the kinetic energy of the hot tea plus the kinetic energy of the cold milk is equal to the kinetic energy of the mixture but on average the tea molecules have less kinetic energy after mixing and the milk molecules have more kinetic energy after mixing.
You are talking about two different cases. At first you say:
We notice that our liquid is no longer as hot as before adding milk.
And your meaning about "our liquid" is the tea.
Then you say:
The total thermal energy of the system (the cup) is conserved.
And your meaning about "system" is tea + milk.
The total thermal energy of tea and milk is same before and after mixing. But, we cannot talk about temperature of the tea by considering to change of thermal energy of both of tea and milk. If we want to determine if temperature of tea increase or decrease, we should consider to its own thermal energy before and after mixing.
In addition, as it has been mentioned in Sammy's answer, you cannot compare temperature of whole of the tea and milk before and after mixing. Because, we cannot define a determined temperature for the system before mixing.
When you say that the system (ie the contents of the cup) is not as 'hot' as before, you are talking about temperature. However, the system was initially at two different temperatures (hot tea, cold milk). There is no obvious way of comparing the two initial temperatures with one final temperature. To decide if there has been a change in temperature, you have to wait until the system has reached thermal equilibrium. Then it is at a single, uniform temperature.
You can of course compare the total thermal energies before and after, to see if there is any loss (or gain!) caused by mixing. This can happen if there is a chemical reaction between the two ingredients. For example, dissolving a salt or sugar in water involves a (reversible) chemical reaction; for some salts there is a change in temperature.