Personally I just use my compost straight off for growing plants in. Annual flowers flourish when planted in it in the summer. However I add everything to the compost; grass clippings, soft growth cut from perennials which have finished flowering, weeds which haven't gone to seed, soft hedge clippings, leaves raked up in the fall, moss and other debris from gutters, kitchen waste, saw dust, emptied out potting compost from flower pots etc. I also add soil every so often to provide bacteria and worms to aid breakdown of the material and if I am digging any holes in the garden, changing layout of beds or edging flower beds, I dump the soil onto the heap. I don't have a compost bin as the volume is so great but this doesn't make any difference as I allow almost a year for everything to decay and start a second heap in the meantime.
Compost aerates the soil, which is essential as it allows roots to spread easily and also provides space for essential oxygen to reach the roots. It retains moisture, and adds bacteria and nutrients. Also it acts as food for worms which in turn aerate the soil. You could mix compost or peat half and half with soil to provide any essential nutrients missing from the compost.