We probably need more information, such as the slope (if any) of the garden and your height above the nearest water course. If the water table is close to the surface I can offer no suggestions other than to build up your beds higher.
To make drainage work, even rudimentary types, you need a fall in the system for the water to run to. One solution is to build a soak-away into the subsoil at your lowest point, 1-1.5 metres deep and a similar length and width which is filled with rubble. Then you have to find a method to encourage the water to run to the soak-away. Underground perforated pipes at the bottom of the top-soil for example.
For the soil itself it is the usual story of tones of organic matter, compost and manure. Grits have a habit of sinking and sand washes out of very wet soil so are less effective.

My garden is basically very heavy wealden clay and the people before me were not gardeners and did nothing to improve it. For many years I have added as much compost, manure and horticultural grit as I could lay my hands on and now the garden soil is easy to work and most plants thrive. When I began, I read somewhere that roses do well on clay and this has been proved true.
It wasn't possible for me to undertake the kind of thing buddlejaga suggests but I think this would be ideal if you can do it.