Answer: Most of your common garden vegetables are going to need supplemental watering in your climate. Except for perhaps some herbs like thyme and sage, and even these won't withstand extreme drought.
I suggest you take some time to improve your soil as much as possible, if it isn't already a rich, loamy soil. Adding organic matter like compost or aged manure will help it retain water. I also suggest burying a soaker hose, if possible, so that when you do need to water, the water reaches the roots rather than evaporating from the surface.
As far as timing, you can plant cool weather crops a week or two before your last frost date (probably the last week of May) but wait to plant heat lovers like tomatoes and peppers until a week or two after the last frost date--say the first or second week of June.
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