When replacing old taps with new ones the most difficult part of the job is likely to be (as with so many plumbing operations) removing the old fittings. Let's first consider wash basin taps.
You must, of course, cut off the hot and cold water supplies to the basin. The best way of doing this will usually be to tie up the float arm of the ball valve supplying the cold water storage cistern so as to prevent water flowing in. Then run the bathroom cold taps until water ceases to flow. Only then open up the hot taps. This will conserve most of the expensively heated water in the hot water storage cylinder.
If you look under the basin you will find that the tails of the taps are connected to the water supply pipes with small, fairly accessible nuts, and that a larger (often inaccessible) back-nut secures the tap to the basin. The nuts of the swivel tap connectors joining the pipes to the taps are usually easily undone with a wrench or spanner of the appropriate size. The back- nuts can be extremely difficult even for professional plumbers!
There are special wrenches and basin or crowsfoot spanners that may help, but they won't perform miracles and ceramic basins can be very easily damaged by heavy handedness. The best course of action is to disconnect the swivel tap connectors and to disconnect the trap from the waste outlet. These are secured by nuts and are easily undone. Then lift the basin off its brackets or hanger and place it upside down on the floor. Apply some penetrating oil to the tap tails and, after allowing a few minutes for it to soak in, tackle the nuts with your wrench or crowsfoot spanner. You'll find they are much more accessible. Hold the tap while you do this to stop it swiveling and damaging the basin.