Why didn't Noah go to Allie's school or home when she didn't reply

The day before Allie left to go home, her and Noah had a fight due to her parents disapproval of him. Noah was well aware of the fact that he was not suitable for Allie as her parents thought and told Allie as much.

Yes, Noah wrote her repeated letters in the hopes that she would write back. Seeing how their last encounter ended, Noah was under the assumption that Allie finally realized that Noah was not suitable for her and moved on with her life.

Noah is introduced in the story as having the type of personality that is non-intrusive. It is shown throughout the movie that although it drives him mad at times, he has accepted the fact that Allie has moved on and he loves her too much to disrupt that.

Also, when Noah spotted Allie in the city and ran after her only to see her having lunch with her fiancee, it drove him into even deeper madness. Before this, he had accepted the fact that he was going to live without her and that seeing her again would only complicate matters even more.

It did not really matter about "strategy". They ended up together regardless, so the way that it happened was the way that it was supposed to be.