Yeah, you should probably focus on leveling and fillout the report cards on the first playthrough. If I had known before starting School Mode, I would have done this approach.
There are two relationship "gauges". One is the Report Card and the other is a heart meter. You can check the heart meter by talking to Monokuma. The Report Card only grows when you spend time and give gifts. The heart meter only grows when you spend Trip Tickets, give good answers, and successfully perform the Trigger Happy Heart.
Keep in mind that there are two consumables that permanently increase stats. Focus on farming items to make these two consumables, fill out your Report Card, and level up your characters on your first playthrough. Ignore Monokuma's requests.
On your second playthrough, you'll have more freedom to pursue relationships. You can max out five relationships per playthrough at most.
I believe you only have to complete the FINAL Monokuma request to "complete" School Mode, but I haven't confirmed this. But you'll need to complete the others to have enough Trip tickets to get five relationships maxed anyway. You must "complete" School Mode to get any of the endings and the 200 coins. To get a specific character's ending, you'll need to have both the Report Card and heart meter maxed and "complete" School Mode.
There is no why! They're a reward in and of themselves.
Another couple of tips: On the first playthrough, you may want dedicated Cleaners so that they level up faster. Keep in mind that the searching activities on the far right side consume more HP. And do NOT ignore the Tools on your second playthrough. They effectively increase your productivity and efficiency throughout the entire playthrough, so if you get enough materials for the Monokuma request, focus on getting the Tools until the next request. You do NOT get any additional bonuses/tickets for completing more than one copy of a Monokuma request.