Personal recommendations:
- Temp controlled solder station. Really, any of them. The Hakko line is fine. Just avoid the no-control straight plug in irons.
- Flux. I like
https://www.amazon.com/AMTECH-NC-559-ASM-TF-No-clean-solder-dispensing/dp/B078W4HBW5/ for the most part. Most any will do. I don't really like the pens, prefer syringe or liquid form (which I use one of the solvent bottles)
- Chisel tips. Avoid conical tips. You want a "reasonable" size one, and a small one for your soldering iron. Dave uses a hoof tip for the drag soldering, which is a great choice but honestly takes a bit more practice to not make a mess of things.
- Straight sharp point small tweezers. There are a lot of variants, something like this works:
https://www.amazon.com/Hakko-3-SA-Non-Magnetic-Microelectronics-Applications/dp/B00FZPEWI6/
- Vision enhancement of your choice. I don't solder with it, only use it to inspect afterwards.
Really nice to have extras:
- Hot air station of some kind. There are some cheap ones on Amazon, can't recommend or not recommend them. Avoid the ones with built in soldering stations, since those are almost always trash. I personally use a Quick 861D now after using the Aoyue [some numbers]++ stations. You don't need anything this nice.
https://www.amazon.com/Quick-861DW-Digital-Station-Display/dp/B00EID23J6. Why hot air? Removing parts. Heat, use tweezers, pop off. Apply flux, heat, place new part on board, bam.
- Boards using a flat finish. ENIG or Silver. Avoid OSP (hard to work with). HASL gets all lumpy and its hard to place chips _flat_ on the board, which matters for finer pitch and higher pincounts and leadless (QFN). Yes this costs more to order ENIG PCBs, but its worth it (plus they have that gold lustre, though cheap ENIG barely contains any gold)
For most chips I tack down one pin on each corner, and then drag solder the rest.
For passive components, I go around the board and place solder on _one_ pad of each component. Apply more flux. Go around, hold the part with tweezers and tack down one side of each resistor/capacitor nicely onto the pad. Solder the second half once you've placed things. Trying to solder both halves at once is much harder, and does require the large chisel tip to bridge both