The solder looks okay. I personally would pass over solder from a China brand because the stuff is easy to buy from others who have been in the game far longer and have established a good reputation, but I have no particular reason to distrust Chinese solder, and haven't looked into how much acceptance it has gained. The main, thing, probably, is that you have made the correct choice by starting off with leaded solder rather than lead-free.
I didn't know the soldering iron at all but looked them up (Tom's hardware) and see nothing to worry about there: they like it.
To run through the process, when the iron is hot enough, you want to clean the tip (damp sponge etc.), tin it, solder the piece, clean the tip, re-tin it, and keep doing that for every single location you solder. I'm sure people bend the rules a little once they're comfortable with the process, but that's best practice, and I think anyone starting out should follow it closely. So even for a capacitor or resistor, you would do a leg at a time, cleaning and re-tinning after soldering the first leg.
As well as the above, you often need to tin a wire you're working on before soldering it in.
When the solder is flowing to where it needs to go, make sure that you don't move the wire within the solder. Give it time to cool and solidify before releasing, otherwise the wire may shift and compromise the joint.
You also want to avoid overheating the terminal/component/trace (that heat will travel far, too), so if you find the solder doesn't seem to take, back off, let the part cool a while, and begin again. If necessary, remove solder so that you can start again with a clean joint. You have to be aware of what's around the location where you apply the iron, as heating for too long can damage or destroy parts, or lift traces from boards.
I don't think it's really difficult, despite the lengthy discussion of tools and techniques you can find even in AK's back pages. More tricky than the soldering itself is stabilizing or accessing the parts in a piece of equipment. Sometimes you're trying to hold a board steady, hold the solder feed, hold a soldering iron, and steady the component you want to solder in to the board all at once. There are various accessories for helping with that, although I haven't had much luck with the third hand tool, and often just use Blu Tack for want of anything better.
