What is Good Insurance Company?

In realistic terms, we cannot claim any insurance company to be a "best" or "worst" at all. But, here I am going to make only the recommendation as per my view. 
While researching for the best insurance, the problem is to decide exactly what the term "best" means. Some clues can be: Cheapest? Pay claims instantly? Great financials? The limited number of policyholder accusations? Or what? Whatever your opinions, the answer tells you what company is "best" for you.
What is Good Insurance Company?
However, here some points to ponder:
One of the most significant points is whether the firm practices self-governing agents, or deals straight with the policyholder. The obstacle with "direct writers" is that their agents (called "captive" agents in the biz) are primarily powerless to assist the policyholder when a dispute arises. The agent may make a great show of "being on your part," but the cookie just doesn't crumble that way. In fact, the best-bound agents would rather be self-governing agents, if they can find an agency whose skills and values match their own. The independent agent has a little further muscle with the company. If a company tightens up on underwriting or claims service, the independent agent can "move the book." That means changing (with the policyholder's approval, of course) all the insureds of one firm to another. On more than one occasion, this is precisely what I have done.
Whether the company is a direct writer or relies on independents, the real agent who handles your account is most important. The agent for a direct writer generally has less reason to be an expert on the subject, because the only thing they have to offer is what their company renders. The independent agency, by contrast, must be savvy about what a variety of companies offer and consequently has an incentive to understand the very subject of insurance, itself. You can, when talking with an agent of either type, discern something about their expertise, in the same way, you develop an opinion of anyone upon whom you rely for advice and assistance.
Don't be influenced by cost. The insurance company especially personal lines (auto, homeowner) is somewhat of a game, in which a company will seek to make a book quickly by offering prices they previously know can't be maintained over time. Take Geico (please). They say you can save "up to 15%." But, compared to what? And you have previously seen the advertising from companies who say something like, "of all those who switched to Auto Heaven Insurance, the average savings was over $350.00." But of course! We're only covering those who switched. But what about those who requested a quote and did not switch? Give us that number.
Then there is Progressive, who seems fixated on the idea that insurance comes in cereal boxes sold by a comedian with greasy hair. They'll show you pricing from other companies, even if the premiums are inferior. You BET they will, and that's because Progressive, like all other carriers, knows what kind of customers they want, and are happy to sluff off the ones they don't want to someone else. So, if they can't make money, then at least they can make someone else lose money.
Price may also vary with coverage. Therefore it is necessary to be sure that quotes are "apples to apples." What are the apples? Just 3 things, basically: liability (usually required by law), physical damage to the vehicle (required by the lender if the car is financed) and coverage required by regulation (usually "no-fault" and "uninsured motorists"). Coverage is one point, the amount of coverage is another. So, when soliciting quotes, be prepared to tell the agent accurately what you want. But first, decide what it really is that you want. In my view, the legally required minimums for liability insurance are way too low, thus asking for a quote on those limits executes no sense. Again, it is just my opinion, but the limit for liability insurance should be not less than the minimum required by the insurance company to hold an "umbrella liability", policy, which adds a million (or much more) to the "underlying" coverage in the auto (and homeowners or renters) coverage. That number is seemingly 300 to 500 thousand per accident. Once you've decided the prudent limits, then ask for quotes based on those limits - absolutely not on the legally required minimums.

This all assumes you really care about what happens when you have a claim; if you don't care, then simply start dialing for money, to find the cheapest deal this year - and do the same with every revision.
Otherwise, a perfectly valid question to ask the agent is, "Over the past 3 or 4 years, how much have your premiums changed, and do you have any intention at all to think the price for your coverage will increase dramatically over the next 2 or 3 years?" See how they handle those questions; look for the "tells."
Here is what an "honest" answer will sound like: "The company I have suggested for you has increased its premiums at an average of roughly 5 to 8% over the past 3 or 4 years. This shows the market cost of litigation and settlements arising from liability claims and the cost of auto repairs. Both of these are rising at more than the overall rate of inflation, but the increases are fairly predictable. I have no reason to think this will change in the near future. I could be wrong, of course, but that's what the record has shown, so far."
Here is what a false answer will sound like: "We didn't get to be the fastest growing (most popular, etc.) insurance company by charging high prices. We deliver a great product at a reasonable price and our reputation and number of policyholders are the evidence that we've been doing right by our customers."
With the above two examples in mind, see which way the needle moves when you chat with an agent.
So, who would I advocate for auto insurance? Although I represented many dozens of organizations in my insurance career of nearly 25 years, I can't confirm any of them as "the best." The honor lies with someone else, but there is a catch. The company is USAA (usaa.com). I have encountered them numerous times in my career, as a competitor and as the company which paid my damage claims against their policyholder. In every instance, I found them to be the professional, expert, courteous and never contentious. The catch? You've got to be "army," or related to someone who is (or was). Their pricing is at least reasonable, and perhaps better. Your mileage will vary, of course, but if you have been or are in the armed co-operation, or are an immediate family member of someone who is, or was, check them out.