A soil can be dry because it is light and gritty or stony and hence well drained, and in this situation addition of manure and compost can do much to improve the situation by aiding water retention. If, however the soil is dry because of nearby trees, then you will probably have to cope by judicious planting unless you want to remove some trees. Mulching is a great way of reducing evaporation of water, but will do little to slow drainage.
Useful trees for this type of situation are acacias and many of the conifers, especially juniper, Lawson's Cypress, and many pines.
Coming down in size to the larger shrubs try olearia macrodonta, eleagnus, callistemon, and abelias. Medium to small sized shrubs would include santolina, hebes, lavenders, nandina, perovskia and osmanthus.
Flowers seem to tolerate dry conditions much better, possibly, being smaller, they require less water than the larger shrubs and trees. So don't avoid anything that you would really like to grow, you can always water a bit at the driest of times. However, the following are perhaps more suited than most to drought. Acanthus and crambe cordifolia, always a spectacular candidates, allium, alstromeria, cistus, euphorbias, echinops, lychnis, gypsophilia, nepeta, gladiolus, salvia, nigella, with its interesting seed heads in autumn, nicotiana (evening perfume), helichrysum, dianthus. There are many more!
Climbers are not so easy, concentrate on jasmine and passion flowers, and try solanum and campsis. These respond especially well to mulching.
For real inspiration do visit Beth Chatto's garden in Essex. Her gravel garden is in one of the driest areas in the country and on an exceptionally free-draining soil and yet she never waters and the garden is always looking in peak condition.