Clay soils aren't the end of the world. The opposite problem, i.e. highly sandy soil is much worse.The problem with overly clay soils, as you probably know, is poor drainage. For some vegetables, like asparagus, they don't like to have their "feet wet."
So as far as watering concerns go, if you do nothing to your soil whatsoever, then you need to water more lightly. The clay will hold on to water much longer than other soil types. You can check your soil moisture levels either by hand or with a relatively inexpensive moisture meter. Just be careful s as not to cause powdery mildew by over-watering.
This is a really simple fix though. Go to your local gardening store and buy compost (or composted manure) or humus and sand (if you live somewhere where sand/gravel isn't readily available). Otherwise visit a local quarry/sand mine
It's a bit of trial and error until you get proper drainage for your crops A simple, unscientific approach is to over-water until you get puddles and watch/time how long it takes for the puddles to be absorbed. If you add too much sand, then the upper layers off soil will be relatively dry. or will fail to produce puddles. When soil is too sandy, then you have the opposite problem (i.e., the soil rarely puddles and drains incredibly fast). In that case you'll notice (e.g. by discolored leaves) that nutrients are quickly washing away and dissipating and are not being taken up by your plants' root systems.
If you happen to amend your soil with too much sand, then you can till in some humus and/or compost. In either case, your heavily clay soil could probably use the addition of humus.
If you don't want to use sand, then an alternative would be to create a potting-type mix. You can add perlite, vermiculite, or any small pebbles to break up the clay soil. Small pebbles can greatly improve drainage.