So if in play, my character spends a lot of time trying to learn a new skill, or a new language, or learn how to use a tool, is there a mechanic for adding that on the sheet? Or does 5E streamline away all such customization with blanket modifiers? I haven't found anything that reflects the ability to grow or develop in a focused way, just blanket improvements as stats increase, proficiency bonus goes up, levels rise, or as multiclassing happens?
The core question is: Is there any focused spot improvement for characters in D&D 5e, or is it all blanket modifier increases?
Most other games have this kind of thing. Take Improv classes? Improve your Performance skills. Go running a lot? Improve your Athletics skills. Practice at the gun range? Improve your firearms skill, all at the cost of some earned resource like XP. D&D seems like characters can sit on the couch between combats and improve as much as other characters who actively work on their skillsets.
Skill/ability is deliberately handled in a very broad/simple way in 5e D&D, and gaining levels is the method by which they are improved
5e D&D represents characters in a deliberately broad and simple way, as one of its design goals is to be easier to play and mathematically less complicated than previous editions.
In that vein, skill and ability is generally represented quite simply.
In general, 5e represents your focused ability with specific skills by determining whether or not you are proficient in their use. If you have proficiency with a skill, it is assumed that as you gain levels, you will be doing the necessary practice and training with that skill in order that you get better at it (as represented by your increasing proficiency bonus).
Certain feats or class features - like the Rogue's Expertise class feature - allow you to double your proficiency bonus with certain skills. This is the closest that the game comes to allowing you represent a specific focus on learning and improving a skill above and beyond your broad competencies. Even then, you will only be able to gain expertise by choosing a class or feat that grants that benefit at an appropriate time when making your character or levelling up; there's no mechanism for gaining proficiency, or expertise, in a skill simply by practising it in play (with the exception that you can learn proficiency with a certain kind of tool or language in downtime).
The upshot of all this is that D&D is not a very granular system for representing complex characters, at least not with player classes. These rules are for representing adventurers; exceptional, multitalented individuals who are typically broadly competent at a wide range of things. It doesn't want an individually detailed representation of precisely how good you are at certain things because it's decided that is overly complex and detracts from the game it is trying to be.
That at least is in contrast to previous editions, specifically 3e - which does feature a much more complex skill system where you assign a number of ranks to individual skills and competency in different skills is much more variable and specific. However, even then, the number of ranks available to you to spend on skills is based on your class and levels, spent when you gain levels, and there is a cap on how good you can be at any given skill based on your level.
In short - you only get better at things by gaining levels, and gaining a level make you better in lots of ways at once. Besides the largely incidental tool proficiencies and learning languages, the system offers no way to represent training and practice improving your abilities - these are things it is simply assumed you are doing, and it pays off when you have sufficient experience to level up.