I just got back into crocheting a couple of months ago. I've been doing charity hats for winter. Long story, but I quit smoking so I looked for something to tuck in my purse and do at break time rather than burn one. I must have looked at a million hat patterns. Anyway, they all crochet flat like your doily up to a certain point where you stop increasing. Whether you do single, half double, or double crochet, the order of crocheting these is always the same. It's a series of staggered increases in every round.
Make a ring, join, work the number of stitches called for, slip stitch to join. The first round is crochet two stitches in each stitch around, join. Chain 1,2, or 3 depending on the stitch you've chosen. Second round, crochet one, two stitches in next stitch, repeat all around. Chain 1, 2, or 3. Third round, crochet two stitches, work two stitches in one, repeat around. Fourth round, crochet three stitches, work two stitches in one, repeat around. Chain 1, 2, or 3. I think you get the idea. You increase one stitch in between until you get the size you want. When you stop increasing and work straight crochets around, your piece will still keep increasing for the next row or two, then it will begin to cup to form straight sides, like for a hat. Hope this helps.
The reason I said "crochet" in each stitch is that you can use any stitch, it will work for single, half, double, and treble. It will help if you use a safety pin or bobby pin to mark the beginning of your rounds.