Without the fuse in, see what your resistance between the 12v side and ground side is on the extra cigarette lighter plug. If it's 0 ohms then somewhere in your wiring you have a wire touching ground. If you do then that's what I'd start looking at.
You said you wired an amp, I'm assuming you ran separate wire for +12v? If you wired it directly to the same line as the cigarette lighter, that's probably your problem... most amps have built-in capacitors for reserve power on instant power draws (bass drum, etc.). Without any resistor to gradually charge the capacitor, there is a high instant draw, which can blow that fuse. I'm not saying you've wired it this way (you seem pretty sensible, i.e. knowing a larger gauge wire is good for an amp), but I've seen some pretty strange wiring jobs and just wanted to cover the bases.
If the amp isn't wired to that circuit, I'd look at any wiring you added like 79Landau mentioned. I've put in a ton of stereos over the years (many just simple aftermarket decks, etc.), and depending on how everything is hooked up - like if the wire is twisted together/taped - you could get a stray strand off the battery lead and it will short to ground even though you thought you've taped everything up nice & tidy. If you have any constant 12v leads loomed/taped together with ground wires, try separating them and see if that solves it. If not I'd start tracing the +12v wire back to its source to see if it's chafed on anything.