Measure the height and width of the window opening, from the outer edges of the casing for the slider windows. Order your new windows based on the measurements. Leave the old windows in until the new ones arrive.
Pry off the border of trim around the inner perimeter of the window opening, using your hammer and prybar. This will expose a space between the edge of the window casing and the frame. Be sure to keep the trim intact as you remove it.
Use your reciprocating saw and metal-cutting blade to cut through the screws holding in the casing, running the blade up and down the space. Pull the whole window unit into the room. Hammer down any remaining pieces of screw in the window frame.
Set your new slider window unit in the opening, bottom first. Push the top back until it comes to rest against the trim border on the outside.
Push shims into the spaces on the sides of the unit while holding your level on the casing. Keep adding shims until the unit is level and plumb.
Drive mounting screws with your screw gun through the sides of the casing and into the surrounding frame, through the provided screw holes.
Spray foam insulation into the gaps around the edges of the window unit.
Use your hammer and trim nails to reinstall the original window trim.
Caulk around the trim and the window unit.