Mark around the starting collar -- it should have tabs around its perimeter -- on the extended plenum, the main duct or the box atop the central air handler.
Drill holes in the corners and several more along the long sides of the collar outline on the plenum side. Use a jigsaw with a metal-cutting blade.
Place the collar in the hole and bend back the tabs against the inner wall of the plenum with pliers. Twist or drill three or four holes through the side of the plenum and a tab. Use an awl or a drill bit smaller than the diameter of your sheet metal screws to create these pilot holes.
Screw the collar and plenum together with sheet metal screws and apply a bead of mastic around the seam, covering the screws. Cut halfway across a strip of fiberglass membrane at 1-inch intervals and wrap the membrane around the collar, splaying the tabs out against the plenum, smoothing it into the mastic all the way around the collar.
Unpack the round duct segments. Separate the segments and slip the seams together in each segment, snapping the hooked sides together along each segment to form a seam. Leave one unassembled for final fitting.
Lay the duct segments out on the floor under the space between the floor joists you plan to use for their run to the floor vent above with crimped ends all pointing in the direction of airflow.
Slide the crimped sides inside the plain ends and secure the segments with sheet metal screws. Apply mastic around the exposed end and wrap with uncut fiberglass membrane.
Tack one end of duct strapping high on a floor joist every 2 to 5 feet with roofing nails. Have a helper hold the duct while you attach one end to the starter collar and attach it as with other segment joints. Raise the duct and nail the strapping to the opposite joists to hold it.
Locate the floor register between 1 and 2 feet from the wall. In the basement, drill a hole in the center of the channel between the joists at least 1 foot from the sill, the foundation for the wall. Upstairs, match the floor grille to the center hole, adjust spacing if necessary, and drill a hole inside each corner to create guides that delineate the parameter for a hole through the floor.
Slide the register boot up through the floor and nail the sides horizontally into the floorboards.
Cut the unassembled duct segment’s length to fit between the register boot and the duct. Roll it together around the crimping and hook the seam while easing it into the bottom of the register boot. Secure both ends with screws, mastic and tape.