The best way to change out a bathroom sink is to bite the bullet and get a vanity top. They don't cost that much more than a drop in or clamp down sick and are a lot easier to install. They also look a lot better too. Here's how to do it and remember that you do any repairs at your own risk. Work carefully and refrain from the hack and slash method of remodeling. You might need the very parts you're destroying. Measure the old top and make sure the new is the same size before you take anything apart.
You're probably going to want to take a long, hard look at the drain and supply lines and if they're chrome, change them out too. Chances are they'll crumble the moment you try and take them apart. The stop valves are also a source of trouble when taking the old vanity top out. If you can't shut off the water with them, get new ones or postpone the work. Working with stop valves that won't hold will only hold up your work and make it much more frustrating. The best time to replace them is when the old vanity top is off and before you install the new one because it gives you more room to work.
It goes without mention that you need to shut off the water before you start working. Remove the plumbing attached to the sink. If it's in good condition, you can reuse it. Just undo the supply lines from the stop valves and the p trap from the tailpiece on the sink. You don't need to use a basin wrench to undo the lines from the faucet; just wait and remove them after the old top it out and only if you intend to reuse the said faucet. Most of the time, anything attached to the old basin is going to be sad shape, but if you can rebuild and clean it up more power to you.
Remove the screws holding the top to the vanity and you should be able to lift it off. I've yet to see a Formica top that's glued on, but there's always a first time. You might have to pry or saw to get the old one off without destroying the vanity. If this is the case, you may have to buy a basin or replace the vanity, which will cost more and take more time to do.
Provided you got the old one off, be sure to dry fit the new one to make sure it fits. Once you're sure it fits, install the tailpiece and faucet to the vanity top and the water supply lines to the faucet tightly (don't wreath on those lines) and tape them out of the way to the top. Dab or add a bead of construction adhesive on the vanity where it will mate to the top. Lay the top to the vanity, letting the adhesive set. The alternative to this is to just connect the plumbing without shifting the top. You can then caulk where the top meets the wall and install the side splash (es) if desired. Run the water and check for leaks. If the tailpiece leaks, take it apart and seal with silicone. Check the water supply lines for leaks and use a basin wrench if you didn't tighten the lines at the faucet enough. You're done. Maranatha!