While, as Jeff suggested, better tools will probably make it easier. I put my board together with a very cheap soldering iron bought at HomeDepot. My solder is also a huge wire and not ideal for electonics.
The trick to getting it soldered correctly, in my experience, is to preheat the pin and pad that you want to solder. Then, when the iron has been touching them for a few seconds, come in and dab the solder to the iron. The solder should practically be sucked down to the pad and around the pin. You will need good lighting and some magnification helps a lot, but I promise you can do it with the equipment you have.
To remove a bad solder can be tricky, but I would suggest getting a cheap solder sucker from amazon. Then, just heat the solder and trigger the sucker on top of it.
Link to solder sucker and wicking braid. No affiliate link, just trying to speed up your searching effort.
