There are a few episodes that go for more dramatic stories beforehand (like the 2nd half of 38, 50 and and the 2nd half of 98), but the first time there's an actual serious arc is the ending of the 2nd season (101-103).
Afterwards, there really isn't an actual continuous serious arc like that again though. Generally, they sprinkle the serious stories between comedic ones even if the serious stories are following up directly from each other, leaving actual continuous serious stories to the movies (Generally the movies were made to be watched between the season transitions, starting between the 2nd season and the 3rd one for the first movie). The movies are kind of mediocre though aside from the 3rd one, which is great.
Also, if you one day make it to the final episodes, keep in mind that the second to last episode is the actual finale. When the show was airing for the first time, it was the final episode shown throughout all of Japan. The final episode number-wise had a limited Tokyo-only late night airing, so it's just an extra story.
Also, two little bits - Starting from the 2nd season, the anime changes from (sometimes loosely) adapting manga stories to mostly original ones with adaptations here and there, due to the manga being monthly. Most people aren't really bothered by that change, but some notice something odd about the show at that point.
3rd season brings a director change on top of that, and the director was a newbie to the position. There's some fun stuff, but several stories there really end up missing the landing. He easily improves throughout the later seasons though, both for humor and serious stories.