This is mostly for PCB/electronics soldering, jewelry soldering is not something I'm familiar with.
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FLUX. Yes I am re-emphasizing this. Now, depending on what you're doing, getting some reasonable quality solder with a flux core can be enough, but having a flux pen or syringe can be good for stubborn components. I keep a pen on hand, though I may move to a syringe at some point. Don't press too hard on the pen, just enough to let the flux ooze out. You may need to shake it downwards before use.
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SOLDER. I use a generic SAC305 lead-free solder, but for beginners a leaded solder is a lot easier (just be careful and clean about it, because you're still exposing yourself to lead, get some lead wipes too). Most importantly, don't use plumbers solder. It has acid to clean up copper pipes that is strong enough to damage circuit board traces and components. Flux is used in electronics solder to clean the oxidation off the copper instead.
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HEAT. A 25W pencil soldering iron is maybe OK for really simple boards without ground/power planes (a plane is an entire layer of a circuit board that's just for the one circuit/voltage). But if you have chonky boi parts like large inductors or any sized part connected to a plane, you'll want more heat. A lot more heat. You can get mid-range 100W soldering irons with a few power levels that could be pretty good for a lot of stuff. Temperature controlled irons are a step up. I personally just upgraded to a JBC Tools CD-1BQF, but that's a $500 soldering iron station. Plus the tips are $40-$50 each (though so far I've just needed two, a third maybe for wipe-soldering.) Is this excessive? Yes. Do I do a fuck ton of circuit board stuff that makes it worth while? Also yes.
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TIP. Make sure you're using the right iron tip. I use a 0.5mm tip for most small SMD components, and a 1.0mm tip for larger ones and ground planes. A larger tip has both more thermal mass and lower thermal resistance. Why not just use a large one all the time? You can damage small parts, it's harder to be precise, and for denser ICs, it's super easy to bridge pins (though flux, again, helps with that) (bridging is connecting two or more pins with a solder blob). If you're soldering some beefy high-current wires, you'll want an even larger tip, or preferably, go to crimp-on connectors (I use Anderson PowerPole connectors and a 3rd party crimper for most power connectors when I can, though if you want ring/spade/flag connectors I recommend the Wirefy crimping tool and their heatshrink connectors. Other kinds of connectors may need different crimping dies. It gets complicated.)
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HOLDER. I really need something better myself. You can get PCB vices for smaller boards. Getting a "helping hands" (a tool with multiple alligator or squeeze-clamp tips) to hold things in place are pretty good and generally fairly cheap. Some come with a magnifying glass (sometimes with a ring light too), but others are just a few hinged clamps on some metal bars with a weighted base. Soldering wires and connectors can be made a lot easier with a holder. Until science can give you another pair of arms, I recommend a holder of some sort.
I could go more in depth about solder stencils and reflow soldering, but uh, that's overkill for most. OTOH, you can get a $200 reflow oven with replacement firmware and it might actually be easier than hand-soldering most stuff these days.