Your choice of this remodeling project is good because it is not a budget-breaker when you do the installation yourself. You should be able to find some attractive white pedestal sinks for under $200 with prices ranging up to about $500. The extra hardware and materials may cost another $20 or so.
Almost all of the major bathroom fixture manufacturers now offer a line of pedestal sinks (the professionals call them pedestal lavatories). Visit a home center or kitchen/bath specialty shop for some style ideas.
You can also contact the manufacturers directly for their product brochures, specifications, and suggested retail prices. Here are some to contact: American Standard, Sterling Plumbing, and Kohler.
Before you start ripping out your old sink and vanity, consider storage space in your bathroom. Obviously, there is no storage room under a pedestal sink. For convenience, you may have to install some wall cabinets for convenient access to frequently used items. This will add cost and may detract from the look that you desire.
Assuming that you have decided to go ahead with this project, you must first remove the old sink and vanity. Turn off the hot and cold water valves under the sink and unscrew the supply lines to the faucet. Put a large pan under the trap and remove it. Some dirty water will run out.
Your children with smaller hands can help here. If the sink has a plastic or ceramic top, it is most likely screwed to the cabinet below. Have a child squeeze under the sink and remove the screws. Lift off the sink. If it is a cultured marble sink, it is probably just glued in place. Place your foot on the cabinet floor and lift up sharply on the sink to break it loose.
The most difficult step for inexperienced do-it-yourselfers is to prepare the wall to accept the sink. Set the sink on the pedestal in its future location. Mark the location of the screw holes recessed in the back of the pedestal where it will be bolted to the wall.
Remove a piece of drywall about 10 inches high by 24 inches long. You will have to install a horizontal two-by-six support piece of lumber across two of the wall studs. The sink will attach to this support piece with screws.
Make a series of 1.5-inch deep saw cuts in the studs and chisel out the wood in the studs. This will create pockets into which the two-by-six support piece will be recessed flush with the studs. Cover the hole with drywall and finish the gap with drywall tape and compound.
Set the pedestal in place again, and place the sink on it. Mark the exact location of mounting holes on the wall (ideally over the support piece) and the location of pedestal mounting holes on the floor.
Move them out of the way, and drill the holes in the wall and floor. It is best to install standard hanger bolts in the wall for the sink and use lag bolts in the floor for the pedestal. When mounting the sink and pedestal, only snug the fastener. Over-tightening can crack the material. Attach the plumbing kit that came with the sink.