Turn off the water supply to the faucet before you begin. Remove the old fixtures entirely, along with any tubes that would get in the way of your measurements.
Measure the relevant distances from the water output of the faucet to the knobs, if they are separate. Mark the center of where each will sit on the counter using a carpenter's pencil if it will show up, or a marker if it will not. Mark the diameter just wider than the diameter of the pipe leading down from the faucet.
Choose the correct bit for your counter top material: a masonry drill bit for granite or other stone countertops, a wood bit for wooden or wood-product counters, a ceramic bit for ceramic counters or a step drill bit for steel or polymers. Use a bit the size of the hole you will make or smaller. Most faucet connections are one inch in diameter or less.
Drill a hole all the way through the countertop to get started if the bit is smaller than the diameter of the hole you will drill. Gently move the drill bit in a circular motion against the sides of the hole until you've widened it to the marks you made on the countertop.
Check that your faucet and connections fit in the hole you've drilled. If it is too small, widen the hole with the drill, working in a circular motion. Check occasionally to see if it's wide enough or if you need to continue widening. If the hole is slightly large, you can fill in the space with caulking or plumbing adhesive from the top and the bottom when you install the faucet itself.
Clean any scraps, bits or dust away from the hole or holes to prevent problems with proper faucet or adhesive fitting.