First remember to have your teenager do the work. Actions bring about consequences and this is a good object lesson. Basically you will need to measure the space in which you want to place the vanity top. If you are doing a one piece cultured top, this is the widest and longest dimension of the whole vanity. There is typically a 1/2"-1" lip or overhang per side, and flush at the back where the splash block comes into contact with the wall. These vanity tops are generally cast in a few standard sizes. Once you know the space available, purchase the top that will best fit these dimensions or get the next smallest size. The cabinetry under the sink should correlate or be just under the dimensions of the top, (in order to provide a lip in frnt and on the sides to prevent water from dripping in between the top and the cabinet.) Again, the cabinets generally come in a predetermined size, but there is a wider range in cabinet sizes than the tops. Once the cabinetry is assembled, (I can be no help here, and generally neither are the instructions) it will need to be leveled and if there is a back piece it will need to have the plumbing trimmed out. The top can be attached by either clamps or a acrylic or latex sealant if there is no mounting hardware. Latex caulkng or sealant is easier to lean up after, and once it dries there is a good bond that water will no longer be able to penetrate. For clean caulk lines, I recommend using the back of a plastic spoon in order to get a nice convex bead. The long handled spoon from Dairy Queen work a treat for this. They are a bit narrower and are highly flexible. I've been able to use the same spoon several times.
Replacing the vanity is certainly something you can do yourself, or rather your daughter, with some supervision. Once you have the dimensions, work people in a hardware shop can make sure that you have the necessary materials for the job. Just don't get hosed on making a lrge tool or equipment purchase if you won't use the tools often enought to pay for themselves. If you know an older individual, they have exerienced things of this nature possibly several times, and are sure to have a collection of tools that would benefit you. All the tools that you will need include:
#2 philips screw driver
a drill (for connecting a top with hardware, fastening the cabinet to the wall studs, or starting a pilot hole if you need to cut for plumbing)
jig saw (for plumbing trim out)
and a lighter
You will need the the screw driver for the cabinet assembly and possibly the hammer as well. The cabinet needs to be level against the flooring, and a level top for the vanity. The caulk gun for the tube of caulk that you will be sold, and the drill for fastening the canity to the wall, or for the other reasons mentioned above. The jig saw as well has been mentioned. There is a good chance it won't even be used. The caulk will beed to be either run along the top edge of the cabinetry which will adhere to the vanity top. Also, a bead of caulk needs to be run along the backsplash and the wall. The spoon works well here in making a nice caulk seal. The lighter I use to burn the lousy assembly instructions that you are bound to receive with the cabinet for your vanity.