To frame a wall you need a bottom plate, a top plate, and the studs to connect them together.
1) Use a chalk line to mark the inside edge of the planned wall onto the concrete basement floor. A chalk line will ensure a long, straight line.
2) Cut 2 pressure treated 2″ x 4″ boards to the length of the wall. These will be the “plates” that form the top and bottom of the wall.
3) Line up the two plates and mark them with a pencil every 16” for the studs. It’s important to do it this way to ensure that the marks are exactly the same on both plates.
4) Lay the first plate along the chalk line and attach it to the floor. Using a hammer drill with a 3/16” masonry bit, make a hole and then use an impact driver to drive in a 3” Tapcon screw. Put one screw at each end of the board and repeat every 16″ along the bottom plate.
5) Attach the top plate to the ceiling joists. If the ceiling joists run perpendicular to the bottom plate, this is pretty easy. If the joists are parallel attach 2″ x 4″ blocking boards between the joists so you have something to nail into. Make sure the top plate is perfectly aligned with the bottom plate before nailing it in place.
6) Measure and cut a 2″ x 4″ board so that it fits comfortably between the bottom and top plates. Insert the stud between the plates. What’s important to remember here is that the CENTER of the stud should be placed on that 16” marked line. Make sure the stud is straight and then attach it to the top and bottom plates using 3″ nails. Repeat for the remaining studs.
7) Now you need to brace the studs with blocking boards. Measure and cut 2″ x 4″ boards so that they fit securely between the wall studs. Attach them to the studs using 3” nails, placing them approximately halfway up the wall (feel free to stagger them in order to make nailing easier).
Don’t put too much stock in trades who give free estimates over the phone. A plumber or electrician can’t properly diagnose an issue unless it’s in person. A free estimate given over the phone is likely to go up significantly once he or she is on-site.