I started with replacing my irons first and got a used set. Then shop for the longer and shorter distance clubs once you know the distances with the irons. My reasoning is that the irons comprise the bulk of the clubs in the bag.
Once you're comfortable with the longest iron then probably a driver, then fill the gaps with woods or hybrids.
Short shots and wedges, see where your iron set gets you, then find what works for you and your course conditions.
This works for...