I personally prefer to build around the wagon with walls. Wood is light and cheap, and the walls can be dismantled later if you want to reclaim the surface and still need wood. I make my walls with a 5x5 interior. Once the walls are up, which takes about a day, you can worry about the rain. Build a ramp to get up to the top level, cover everything with flooring spare for the tile right above the ramp. This takes about another day. Once the roof is finished, deconstruct the ramp, and build a wall where it was to hole up the last tile of roof. You now have a complete seal from the outside world in three days, and all your stuff is safe. I still suggest moving your supplies underground with a dump zone in a moderate amount of time, but that can be done at your leisure.
Using this method of securing the fortress, I embark with two miners. The miners can start digging right away while the other five build the walls, and later the miners can be drafted into the military, which is normally done when the first migrants arrive. The miners that I bring do not have any mining skill and instead have soldiering skills. Casserite, malachite, and a skilled arms/armorsmith are brought to supplement this, so a functional and equipped military is ready to be created by the start of the first summer. With a military in existence, the caverns can be breached with some degree of safety for the purpose of agriculture (if you're on a biome with no soil) and the pack animals can be butchered without fear of their skins causing injury.