The easiest way to solder multi pin connector?

https://www.parts-express.com/Desoldering-Pump-370-030

Heat, de-solder and suck the old solder off.

I use a sharp round pick after I pull the solder off. I insert it and make sure the hole is de-burred. Clean the pins or wires with a little isopropyl, insert them through the cleaned up holes and little dab of flux and solder. Clean afterwards with isopropyl alcohol. Check your work. If there is any issues, repeat the process. 

It will take longer to heat up the soldering iron than to do the work. :-)

Regards

De-solder braid is just fine. Clean like Dave said. 

Most important with boards like this is to use the right temp and right size solder tip and even then be especially careful not to overheat. Because your biggest risk on a PC board is overheat and lift or burn the trace. Especially with a lot of them close together like this. 

First, learn to solder. I bet there are some videos on YouTube. Then get the right equipment and practice until you get the hang of it. 

@russ69 ,

I’d second that. There’s nothing like practice when it comes to soldering.

Cleanliness is crucial. Otherwise the solder doesn’t stick well.

I used isopropyl alcohol when replacing the 3.5mm jack on my Sennheiser headphones.

Here's the first one of a nice video series on YouTube I found both helpful and informative.