To add to the wisdom of Doc B, the only components that you could possibly damage from excess soldering are the LEDs. They don't need a ton of heat/solder for a proper joint, but feel free to cook the rest of the parts as much as you need to get the solder to flow into the board.
For each board hole, there's actually plating in the hole, so the solder will eventually flow down into these plated holes and that's how you'll end up with really good joints.
