The process for filing a homeowner insurance claim is not overly complicated, but you'll want to make sure that you don't miss any of the following steps:

Contact the Police to Report any Crimes. If your claim is the result of theft or other criminal activity (such as vandalism), your first step should be to contact the police to report the crime. Keep a copy of the police report as confirmation and share it with your insurance company.
Contact Your Insurance Company. Reach out to your home insurance provider immediately. If you have any questions regarding coverage, costs, deductibles, etc., an insurance representative should be able to walk you through your issues and inform you of any other next steps you should be aware of. Your insurer will send you claim forms to fill out, and will likely also assign you an adjuster to inspect the damage.
Document the Situation. The more clearly you can document damages and losses, the more accurate your claim will be. Take detailed photos and videos of any areas of your property that may have been affected, and (if possible) include images of those areas prior to the event. If you have visual records of any damaged or stolen items, include those as well.
Make Urgent Repairs. If your property is at risk from further damage, make any necessary, urgent repairs while you wait for the insurance adjuster. This may include basic, stopgap solutions such as covering a damaged roof with a tarp, or more in-depth repairs that require additional costs. Keep your receipts for any repairs you make; your insurer may be willing to reimburse you.
Fill Out Claim Forms. By this point, you should have received the claim forms from your insurance company. Fill these out completely and accurately and return them immediately to help move the process forward. If you have any questions about the forms, don't hesitate to contact your insurance provider.
Work with the Insurance Adjuster. If an adjuster is sent to your home, cooperate with them and give them access to any damaged areas they may need to inspect, and have a list of damaged items ready and available. The adjuster may also wish to interview you to get a more accurate idea of the situation; answer their questions as thoroughly as you can.
Keep All of Your Receipts. If you don't have the right documentation, then your insurance will not be able to reimburse you for your expenses. Keep receipts for any urgent repairs and living expenses that are covered in your policy (such as hotel rooms). Once a contractor gets involved, ask them to make their invoice as detailed as possible, and to itemize expenses such as labor costs and materials. If you are reporting a liability claim, write out a description of the events that took place, and have a witness verify your account (if possible).
Document Conversations and Interactions. Seeing your insurance claim through to the end will likely involve a number of emails, telephone calls, and other interactions. Document these conversations and record when each took place, what details were covered, and the names of anyone involved (insurance representatives, police officers, etc.). Communicating over email allows you to save and review entire conversations, making it easy to find specific information when you need it.
Regularly Follow Up. Provided everything is in order, there's no reason why your provider shouldn't be able to resolve the issue quickly. Keep on top of your claim by following up with your insurance company on a weekly basis. Additionally, make sure that you are keeping track of any deadlines or responsibilities on your end.
Review Your Settlement. Once everything is finalized, your insurance provider will send you a document detailing the settlement offer. This will specify the terms of the settlement. If you are not satisfied with the offer, you can contact your insurer to have them review the claim or consider new evidence or other documentation. You may also wish to hire a third-party public adjuster to help you contest the claim. Be aware that once you accept the settlement, the claim is legally closed and cannot be contested further, so make sure you review it closely before you agree.