If I got in to a car accident with an insured, what is the next step to get a fair compensation from my insurance company?

From your insurer: you can submit a claim to your insurer and they should pay (regardless of fault) IF you have the appropriate coverage. However, auto insurance policies tend to NOT have coverage for the personal injuries or medical bills of the insured--unless you had some sort of policy which specifically covered you for your own injuries, your insurer would not have to pay you anything. Insurance policies are contracts; like any contracts, the other side only has to do or pay what the specific terms of the policy require them to. So your first step is to review you insurance polices and see if they provide you the relevant coverage; then if they do, you can submit a claim under them. 
If you don't have the relevant coverage, you could try submitting a claim to the other driver's insurer, but they are not obligated to pay you; they will pay only if the other driver has applicable insurance and the insurer believes that under the circumstances, it is in the best interests of their insured and themselves to pay. If the other driver does not have the applicable insurance or his/her insurer chooses to not voluntarily pay you, you would have to sue the other driver for compensation. But since the compensation you can get is equivalent to your out-of-pocket medical bills and for long lasting disability or serious life impairment, some amount of "pain and suffering," it is not at all clear suing for an ER and an urgent care visit and a "little bit of pain" would be worth the time, effort, and cost of litigation.