Electronic soldering and repair for a newbie
Get some solder wicking braid if you really want to mess with removing the old solder.

Otherwise, just re-use the old solder by melting it, pulling out the old, snapped-off wire bits, then insert the new wire ends through the molten solder pools. 

As a more important tip, buy a quality soldering iron/station.


If you're not sure how committed you are to doing soldering/EE work, then you can buy something like a Hakko FX600 iron for ~$40 that will do a decent job (anything that costs less, isn't worth throwing away, IMO).  
If you're more sure of your commitment, something more like an FX888D is much better, and your soldering abilities are likely to improve faster while learning with it!