Which are the steps followed in taking up an insurance policy?

Let's start with the basics.  An insurance policy is a contract between an insurer and an insured under which the insurer promises to pay a determinable amount to the beneficiary(ies) of the policy if one or more events occur in the future. 

In order for the contract to not be considered gambling, the person taking out the insurance needs to have an "insurable interest" in the item, person, or company being insured.  For example, you can't buy auto insurance which pays you money if your neighbor's car is damaged in an accident.  Likewise, you can't take out life insurance on someone whose death would not negatively affect you (e.g. total strangers).

The contract needs to have a way to set a specific amount on the payment.  For typical personal and corporate policies, this is some portion of the value of some physical asset, like the value of a car or truck or building.  It can also be meant to pay amounts for which you are determined to be liable by a court of law (liability insurance, "personal umbrella" policies, etc).  There will usually be a deductible, which is an amount of loss for which you are responsible before the insurance policy pays anything.

Finally, the contract can only be for events that have not yet happened, and which you are not planning on having happen.  For example, you can't get a new insurance policy for your car to cover a crash that happened yesterday.  Also, if you take out insurance and it is later determined that you intentionally caused the event leading to a payout, typically the contract becomes unenforceable.

The insurance company will generally go through a process known as underwriting to assure all of these factors are met, and to determine the price at which they are willing to offer you a policy.

That being said, your steps break down into:

1) Determine what sorts of events you want to cover against (car crashes, being sued, a relative passing away, etc.).

2) Determine how much compensation you would like in that event (subject to limitations as described in the background).

3) Assemble the information an insurer will want for its underwriting process.

4) Contact several insurers, either directly or through a broker.

5) Submit applications, compare the offers they give you, and select the one that has the best combination of terms and premium costs.

It is important to note that buying insurance, for either personal or business risks, should be considered as part of an overall financial and risk management plan.  This is the most important part of steps 1 and 2.  It is common for people to underinsure to save premium costs when they have limited income and financial assets, although arguably this is the circumstance in which they need insurance the most due to lacking the assets to deal with the event themselves.  For example, if you depend on your car for getting to work, and you don't have enough cash to buy a new one if it gets wrecked, you should make sure you purchase adequate collision coverage so you don't end up carless.

Finally, note that insurance products can become complicated, with differing combinations of deductibles, benefits and premiums.  The can often differ as well in the restrictions they place on the situations generating the loss.  It ultimately helps to (a) work with an agent you can trust, and (b) be ready to lay out the policy features in some kind of spreadsheet to analyze the differences and make a good decision.