How to judge if a reaction is exothermic or endothermic
This is one of the most commonly misunderstood principles in thermochemistry - if you can wrap your mind around it, you are most of the way there!

The good news is, there is an easy way to do that. In any physics problem (thermochemistry is really physics) you have to be very careful with how you define the system and the surroundings. This is important for two reasons.


It lets you write energy balance equations across the system boundaries (so we can use math to solve the problem)
By choosing the right system boundaries, we can make the math much, much easier.


Since thermochemistry at this level always involves the same types of problems, chemists have done the hard part for you and already selected the best system boundaries.

The system is whatever is in the reaction equation.

That's it. The entire rest of the universe is the "surroundings."

Looking back at your question:


  The thermometer is in the solution which means it measures the temp of the solution. It's the system. The temp of a system increases means heat flows in, how could this be an exothermic reaction?


If the reaction molecules are the only thing in the system, then that means the water, the calorimeter, the themometer, you, me, the room - everything else - counts as surroundings. Now, the water did heat up. We know this because the thermometer absorbed some heat and showed a temperature increase. Since they are all part of the surroundings, the heat must have come from the system. Since heat left the system, the sign of $q_{reaction}$ is negative, and the reaction is exothermic.

As a teacher, I have found that getting system boundaries and the signs right is one of the biggest problems students have. If you focus on getting that concept down early - and really understanding why they are that way, you are well on your way.