It will be easier to solder with old-fashioned eutectic solder (containing lead and tin) if you can find it, especially if it has a core of rosin flux. However, I'm trying to avoid lead altogether, so I selected lead-free (silver containing in this case) solder, and an additional source of flux in a pen, if I'm having difficulty getting it to flow or wet.

I also tend to solder things like this with a high temperature (400 DegC) as it means that re-flowing joints and removing solder bridges is very quick, even when using copper solder braid. Just keep the heat on for a bare-minimum amount of time so you don't damage the glue that sticks the copper to the substrate (or melt the plastic parts of the headers!)

As might be visible in the images, I'm using a chisel shaped tip on my iron. This is the most appropriate type for this through-hole soldering.

You should checkout the datasheet for the solder you have and select an appropriate temperature!

When you heat up the iron, apply liberal solder until it is completely covered in melt, before wiping to remove any crust, re-apply solder a second time, and park it in the holder. Always apply solder to the tip immediately before placing in the holder between uses! This is especially important with lead-free solder, as failure to do this will quickly result in an oxidized tip which will most likely need replacing - cleaning is possible with fine wire wool, but usually it is too late by then.

You can buy 'solder tip resurfacing' kits - these are little pots of solder plus flux and can sometimes rescue a bad tip.

You've got a few spares anyway right?