How long does it take to get health insurance coverage?
If you are an independent buyer (not getting the insurance from work), it is generally a fast and easy process (especially if you are young and healthy) and will be in effect as soon as the paperwork is processed (about a few days with most insurance companies).
If you are getting insurance through your employer, as soon as the employer offers you insurance through the company and your paperwork is filled out, it is effective immediately. Most employers have minimum requirements for your being at work before you are offered company benefits such as insurance (you might have a review period/probation period or other full-time requirements).
It depends on your circumstances and the chance that you might be rejected for pre-existing health conditions. There are laws preventing insurance companies from discriminating against certain conditions, but you for the most part just need to review your options and what plan works best for you. The rest is simply a matter of waiting for approvals.