The best time to plant your annuals would be towards the last frost before the start of spring.
That said you need to know that annual are very hardy and can survive most climates except severe cold. So avoid frost. If you have long summers, it's great.
The two common times people go in for annuals are as below
Since most annuals are sensitive to extreme to medium cold, it makes sense to wait for the frost to subside before planting your annuals. Make sure the soil is no longer chilling or cold before you plant your annuals. Warm soil works best in these cases.
Few annuals you can consider planting by late spring are as below
Cleome
Impatiens
Petunias
Beginning of summer may be the right time to plant annuals which thrive in hot sun. There are several annuals which are drought resistant and don't mind direct sun hitting them. If summers start early in your region, consider planting the below annuals
Marigold
California poppy
Zinnia
If you consider planting other annuals towards end of spring or beginning of summer, it should not be a problem. Just ensure that the summer heat is not intolerable.
You don't want the annuals to go into a shock. If the soil is very hot try provide an artificial shade or plant the annuals in areas which are naturally in shade.
The annuals don't make a lot of demands as far as watering is concerned. To ensure healthy growth however, you should provide ample watering.
Annuals are usually low care but don't neglect them. Ensure that you soak them deeply at least twice a week.
Watering regularly would ensure that the stems stay turgid. There are more blooms on a low stressed plant. An important tip to follow is to water in the morning before sunrise if possible.
Avoid watering your annuals towards evenings. The lack of sun would usually cause water logging and lead to diseases like mildew.