Buy lots and lots (and lots) of sacks of cheap multi-purpose compost (peat-free).
Don't bother trying to pick or move rubble out of the clay, unless it is chunks on the surface that you can just pick up and bin as you go.
If there is no grass in the area, just spread two or three inches of compost all over the new space intended to be the bed, just so it looks nicer, like a proper flowerbed. Even if you don't want raised beds, get some decorative edging to delineate bed from (future) lawn.
Buy plants from a garden centre for instant results. Then when you plant each plant, dig a hole bigger and deeper than needed, and line it with compost before putting the plant in and filling with lots more compost.
The worms will work the compost into the soil. Every winter for the next few years put another couple of inches thick of compost all over the beds.
If there is rough grass or weeds, that is a bigger issue. Cover the area with sheets of cardboard (from cardboard boxes cut up), then cover with three inches of compost. Cut a hole in the cardboard to plant each plant, and line the dug hole with compost as before. The cardboard stops light getting to the weeds so they should die off, and eventually the cardboard just rots away.
Be prepared to spend time weeding - if lots of little weeds appear on the compost, rake them to kill them off before they get too big to tackle.
Expect some of your new perennial plants to die - unless you do lots research in advance to find out what suits where, and even then you could get unlucky. So don't worry if things die. (Annuals die every year of course).
You could just sprinkle a load of annual seeds in a corner - rake the ground a bit to loosen it up first - and see what comes up.