The specific date your new health coverage starts depends on your particular insurance company, the plan you choose, and when and how you apply.
In general, if you apply between the 1st and 15th of the current month, your start date is the 1st of the next month. If you apply after the 15th of the current month, your start date is usually the 1st of the month after the next. To speed up the process, it's a good idea to e-sign your application and make your payment as soon as possible.
Be aware that during the yearly Open Enrollment Period, it may take longer for the insurance company to process your health-care application. But, as long as you submit your application before the deadline and make your initial premium payment, the insurance company will typically honor your requested coverage start date.