Contact a local insurance broker that works with many insurance companies. A good one will know the market inside and out and will be able to get you the best deal, usually better than you can get by walking into a State Farm office or whatever. They may encourage you to move your homeowners or renters insurance to them as well as often there is a multi-line discount.
I've been *very* pleased with my agent. She answers my questions very quickly and accurately, and when I complain about the rate she does the shopping for me and/or figures out things to change to get a better value.
There is nothing to stop you from talking with two or three brokers, as well as calling up Geico etc. Just be aware that insurance is complex and it's hard to make apples-to-apples comparisons.
I just want to chime in here and say that YMMV with brokers - my wife and I originally had our first car insurance policy with a broker who was very nice, but ultimately wasn't able to secure us the best deals that we were able to find (with decent insurance companies). I think we ended up saving ~$10 a month by 'going it on our own' with our renter's policy. Predictably, our rates started to go up and she had less wiggle room to try and find ways to save without completely sacrificing her commission.
We left that brokerage and started just purchasing 'directly' through the insurance companies' websites. 6 months later, we received solicitations from her brokerage and another, encouraging us to come back, but the rates they were offering were ~15% higher than what we were able to do on our own.
This might just be state-specific, or it could have been because of the specific policies of those brokers - I really don't know. When we purchased our house 2 years ago, we did end up going through a brokerage for our homeowner's policy, one because the brokerage offered underwriting through a local property insurer that's known to be very good, but primarily because we receive (and will continue to do so indefinitely) a discount for using this particular brokerage as part of the purchase process through our Realtor. If this hadn't been the case, I wouldn't have gone with a brokerage.
The other thing to consider, and it's definitely worth considering for folks on these forums, is that when you take your policy through a brokerage you typically cede control of your policy to your agent. For some, this is fine, but for others (like me) who want as much control over their coverages as possible, it's a big drawback. If I want to raise the deductibles on my cars on a whim on a Saturday evening, I'd like to do so without 1) waiting for the office to be open and 2) being lectured on why that's inadvisable. It already irks me enough that I don't have that flexibility on our homeowner's policy and it's enough of a bother that despite the discount, I do see myself eventually dropping the broker and just handling it myself.
I will also admit, that I generally do not find any enjoyment in working with salespeople (other than in a professional setting), and I tend to view insurance brokers as salespeople first and foremost.
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To the original question - our rates with GEICO went up pretty much identically. We did have an accident (to which I admitted fault to GEICO - it was indisputably my fault) which may have played into that, but I do agree with others here that insurance is probably just going to go up in general. Switching will save you money, but there's little reward for loyalty these days with the national insurers. I was speaking with a phone agent from our previous insurer (Nationwide) a few years ago who even admitted that there's no reward for sticking with a particular insurer for more than a few years - at least when it comes to the national ones.
I'd encourage you, in your search, to find out which companies will have better customer support and more comprehensive coverage where you live. There are national trends, and there are local trends as well. Across all my interactions with insurers, the most impactful positive experiences come from companies who have agents and customer support that are prompt, helpful, and willing to work with you in a given situation.