What weapons to upgrade
In this guide, I will mainly focus on the first half of The Last of Us Part 2, as things go on for about 10 hours or so. The loading of your weapon changes a bit throughout the game, but especially here. While I am focusing on the first half with Ellie, the tips carry over to the back part of the experience as well, so just change what I say or else I'll mess up a lot. And if you need to know how to upgrade your weapons, check out this guide.
At the end of your time with The Last of Us Part 2, your weapon collection will have increased a bit. If you aren't planning on launching a new round of play then the first tip I have for upgrading weapons is to focus on the ones you start the game with or acquire a bit quickly. While many of the latest weapons are very useful, you won't be using them for very long.
This means that you should try to focus on your pistols, especially the semi-automatic, and your rifle. These are both useful and with you most of the time, but two weapons you get later are so powerful that you'll want to upgrade them in a few key ways as well. These are the shotgun and the bow (see where to get the bow here). What you focus on should really depend on your playstyle and how many games you have on hand. Let's discuss.
Stealth players trying to take out enemies from afar or silently up close will want to pump upgrades in the Semi-Auto Pistol and Bow, with the Rifle a third option thanks to its 6x range upgrade. The arc is silent all the time, and the semi-automatic pistol allows the use of silencers once you gain the ability through upgradeable skills. The major upgrades to be obtained are those that add stability to your aiming and the range finder to the bow. It helps a lot with targeting enemies, especially getting all those important head shots.
If stealth fails you more often than not, the shotgun is your best friend, with the shotgun as a solid backup as well. For the shotgun, you can improve its reload speed and damage, both of which are very useful against larger enemies such as shamblers. And all of that translates into the second half of the game, focusing on silent weapons for stealth players but upgrading to more powerful (and stronger) weapons if you're more of a flamboyant style player.
In short, if you are asking which weapons to improve in The Last of Us Part 2, the answer is which one you will use the most in your particular playstyle. Once you decide you can fine tune specific upgrades for your limited parts. Stability and aim are best to hit first, with ranges and other options to come.