As a rule of thumb, first playthrough don't really worry about any of these questions. Find a comfortable weapon and put your souls/echoes into vitality, endurance, and a stat that compliments your weapon of choice (more than likely either strength or skill with early access weapons in Bloodborne)

Now, in regards to your questions I will answer as if this isn't your first rodeo in Bloodborne. These are my habits in these games, not necessarily the best haha

-How do you approach making new characters?

I usually pick the class that starts at the lowest Soul/Blood Level. This tends to maximize the custimaztion you can get. I can't remember what the class is in Bloodborne, but the Pyromancer in Dark Souls starts at SL 1.

-Equipment is quite spread out across the different worlds. Do you find yourself hopping from world to world?

Your build should be built a weapon you want to maximize output for, so assuming you've beaten the game and have a good idea what you want to use and where to get it, you should have a good idea of the fastest path to get it. In Bloodborne, it really depends on if you can get into a Chalice dungeon that has it before it's available in the main game.

-How do you feel about the locations (specifically the locations) of weapon upgrades?

I'm not sure I know what you mean by "weapon upgrades". You could either be talking about the blood shards/chunks/rock that level up your weapon or the gems you put on your weapon to add stats/effects. Chalice dungeons people have shared online are again a really good resource for these, the gems more than the upgrade shards. Again, once you get familiar with where the important ones are, you usually don't leave the critical path unless it's really worth your time and can benefit the build. The gems found in the main game are more than adequate for playing through the game, just equip the ones that maximize damage output. However, entering the world of PvP, high quality gems from Chalice dungeons are essential.