On a basic level, everything in the game is missable. You only have a set number of days until the end of the game, so you’ll need to do your best to balance out your adventuring, skill development and social links.
By passing these up, you may not max out this skill by the end of the game, or have it high enough to trigger certain events/social links as you go through the game. However, there will almost always be skill point activities available in one form or another until the endgame.
Persona 4 Golden added two new social links that must reach a certain level by a certain date before they actually are missed.
The names and dates are:
Marie must be completed by 12/23.
Adachi must be level 6 by 11/01.
Trying not to spoil too much, these new social links are semi-important and count towards unlocking the additional content added in Golden. The max level of these links go up as the story progresses, so don’t worry if it seems like you can’t progress.
In both versions of the game:
Dojima and Nanako can only be completed before the start of November or so.
All the others can be missed in the sense of running out of time before the end of the game, but in essence their characters will remain available (according to their schedule) for the majority of the game.
With the facts out of the way, I feel like I have to editorialise a bit - as you’ve guessed, this is definitely a game you’ll enjoy more if you don't worry too much about missing stuff or maxing everything out. While it is possible to get everything in the game in one playthrough, it involves adhering to a fairly strict schedule that you’re very unlikely to hit on by chance.
So try not to worry too much, (with the exception of the two P4G characters mentioned above) nothing is critical and you’ll still get a rich experience out of the game whether you hit everything or not. If you permanently miss something, there’s always the New Game+ or even YouTube - so enjoy yourself!