Good idea, wreak maximum destruction asap. Hmmm...
Or would you think it might be better to think on what you would like to have, and work towards it. You'll probably find your ideas change as you progress, so you would have the flexibility to adapt your design.
I'd deal with the existing concrete bits first, as the grass can look after itself. It might look untidy to some people, but there are certainly things living in it that are quite happy to be there.
I would start with the lawn because for not a great deal of time, money and work, you can turn something ugly into something beautiful quite easily. Good project for a beginner. When you stand back and look at it knowing you did it yourself its a good feeling. Gets you into it.
Its looks are a matter of opinion only. There is no absolute as to whether it is "ugly" or not. The fact remains there will be wildlife living in it now that is quite happy with the situation. There is not much lives in a neat lawn.
I would agree the first six feet could look more appealing. Removing the debris and the grass etc from that bit and planting something or turfing might be an improvement. I'd perhaps mow a wiggly/meandering (anything but straight). path down to the end and a small area down there. It doesnt look like a bad mix of grasses, and If you had to turn it into a lawn, mowing and raking off might be all that's needed.
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