Thanks! I'm guessing the power adapter is US?
I know i should not buy a really cheap one, seems important to have a good one!
It is important to get a good one, but it doesn't have to be expensive.
While Weller is a good name brand (If I were outfitting a professional lab, I'd lean that way for the "I can't afford for it to break" reliability) IMO, it's a bit too expensive for fine soldering for the hobbyist. While that $40 Weller Iron is a good deal for what it is, I'd also mention it's not thermally controlled, it's power controlled, so for what you're wanting to avoid burning out boards/parts, it may be a finicky tool.
A thermally controlled iron has a sensor in the iron constantly tracking the temperature of the iron and adjusting the power output on the fly. Sitting in the stand it will draw one power level. Touch it to something to solder, as the temp drops and the iron automatically cranks up to the needed power to keep the temp at the set point.
This means you can have a 100W+ iron capable of reaching temperatures that melt through the printed circuit board if left to run at full power . . . but if you set temp is to just above the solder's melting point, the iron never gets any hotter than that temp. Better still, with the lower soldering temp (backed by the "when I need it" power) a VAST majority of the parts you solder around can now tolerate the temp -- you won't burn out pads on the PCB, and it becomes MUCH harder to burn out components.
So if that iron is unsuitable since it doesn't maintain a safe temp for you, what's a better option?
dunno about you local plugs, but look at this one:
http://www.hobbyking.com/hobbyking/store/__19241__Soldering_Station_with_Adjustable_Heat_Range_with_EU_Plug.html
It's a knockoff of a Hakko 936 (an older but reliable iron, and Hakko is a good brand), but it's a good iron for the price. It's a fairly good copy of a solid design, and the corners YiHua cut are fair. 60W thermally controlled should be MORE than enough for most any fine soldering safely, with the power to do medium sized battery wires with a touch of patience.
One caveat I'd make for the YiHua 936 -- buy an original Hakko iron tip. The soldering tip that comes with it is cheap, but all soldering iron tips are intended to be replaceable. Plan ahead to replace the original tip, and buy an genuine tip from Hakko when you order the iron.