There are 2 main parts to this - do you want to re-design the layout and also what you can do to improve your soil.
The layout question could be solved with having someone do it for you, if it's affordable, as it will be back breaking work - some of us love this though, so it's a fun challenge. I'd look at what you could do with some retaining walls, and largely aim to design around the layout as a feature. Some plants prefer hills etc.
For soil improvement, your best approach is to either add a drainage system - good if you are moving a lot of the soil and/or incorporate lots of organic matter into the soil. This helps make the soil more crumbly, less prone to baking into a concrete like structure in dry weather, as well as less sticky. I'd improve the drainage some what, with or without a drain system, and add progressively as much organic material as you can. This can be composted material or animal manure, preferably anything from a sustainable source, as peat is from peat bogs that have taken thousands of years to develop. Local authorities where I live sell composted material at fairly low prices. For under the lawn, you could also add some sharp sand, that will help with drainage - shady lawns, with poor drainage typically get moss problems, so this is best to plan a solution to prevent this. Lime is also able to flocculate clay soil, into larger particles - this does make the soil alkaline, and thus no good for acid loving plants - it's better overall to incorporate the organic matter, as I think you understand from your question.
I'd perhaps also let the soil drain a bit before embarking on a soil shifting exercise, as when it's drier it will be somewhat easier to work on.
Add more organic matter each year - the earthworms will take it down into the soil. Mulches around plants will decay, and can be renewed too. Choose plants that will flourish in your area, and soil type too, so that you're not fighting a losing battle. Start composting as soon as you can, so that you've got your own free source of material to add in future.
Hope this helps. Good luck! Rob