While I generally agree that a rework station is a great tool for the soldering arsenal, I know you’ve had some trouble sourcing things, and you’ve already spent a bit of money as a result of the recommendations so far.
Unless the rework station really interests you and you have the money to spend on it, perhaps see how you get on with the new tools at hand, first.
FWIW (and this is my 54’th year of using a soldering iron) a 1980’s style Weller TCP soldering iron, (with number 7 and 8 bits) some oldschool lead/tin multicore solder, a tooth brush plus rubbing alcohol, and a solder sucker will do pretty much everything.
I’d heat the pad carefully whilst adding a little solder and use the tip of the iron to rub (gently) at the gold stuff until the solder starts to bond to it. when you have a blob formed, then suck it off with the solder sucker, which will reveal how well you have tinned the pad. if eh flus builds up excessivley that’s where teh tooth brush and rubbing alcohol come in…
When you are happy with the pad, tin the wire, then solder the two together.
what solder are you using? if you trying to use lead free solder it would explain all your issues.
With the new iron, tips and flux. I have no trouble tinning the wires or pads. What I struggle with now is that when I place the wire on top of the pad, and try to solder it. Since the wire covers up almost the whole pad. Solder won’t flow down to the pad. What do I do here?
Edit: finally did it. Had to skip a pad, but I applied pressure with the iron and it melted right into the pad.
Have you tried to change the soldering wire. When the soldering wire is not up to the standards this hapens. You need to change the soldering wire and see before changing the soldering iorn