I'm no soldering wizard, but you definitely want a thin-tip on your soldering iron, and a proper electrical solder. I've only used one soldering gun that was any good, most took too long to heat up and this negated the advantage of the gun. I was quite pleased the first time I used an iron - it was so much easier to control.
Most of my soldering has been on car wiring, and some basic electronics, nothing terribly small or delicate, though I have soldered wires to circuit boards (my only foray into digital, I put USB ports on a few of those "single-use" digital cameras).
This may sound obvious, but be certain you are using solder meant for electrical connections. Once I had one hell of a time trying to solder some wires without melting everything nearby. My father assured me his solder was the correct stuff. I later learned he had handed me solder for plumbing - the ingredients melted at very high temperatures.