Is there a way to get an internet connection for free?

We want to know if there is way to connect to the internet without having to pay the high fees that our cable company charges for internet access ($55 a month). We are on a fixed income and in our sixties. We enjoy being connected to our friends thru the internet and playing games like Mah-Jong and Farmville on Facebook. But when it comes right down to it, when being forced to choose between our cable and internet connection (which together cost $115 a month), the internet would be the one that we'd have to cut, as without cable, we receive no television signals. We are homebodies and love each other a lot (married 42 years), but we need the entertainment factor that watching TV and being on the internet provide us. ;-) We read today's edition of your newsletter about a person asking if they could share the password for their router with their neighbor. We think that we got the gist of what you were saying and wonder... Can this “router thing” allow us to connect to the internet without having to go through a separate company? How much does a router cost? Is it hard to hook up? Is it something that we could do ourselves?
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A router is not going to get you free internet, no.
Besides, you probably already have one.
I like to compare internet service to your landline phone service: if you want your own phone, you gotta pay for the connection. Without that, all that you can really do is borrow your neighbor's phone or maybe that of a generous business.
But if you want your own connection at home, that's a service that needs to be paid for, I'm afraid.
I'll look at a couple of alternatives for internet, both free and paid, and talk a little more about that router.
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About that router
A router is simply a device that allows you to share an existing internet connection.
"The options for getting no-cost or low-cost internet into the home are few."
Because you have internet right now through your cable company, it's very possible that you already have a router, particularly if you have more than one computer. Whether or not you have one, however, a router is simply a device that is used to manage the internet connection that you already have.
In the article about the person sharing her connection with her neighbors, she already had a router connected to her internet service (that she was paying for) and was allowing her neighbors to connect to it.
The two basic types of home internet service
Internet service to the home is often one of either two types of service:

Cable - Cable internet makes use of the existing cable coming into your home provided by your cable TV provider. They use that same cable to bring in internet service. In this case, the cable company provides your internet service.
DSL - DSL makes use of the existing telephone connection into your home provided by your telephone company. They then use the same wires to provide internet service. In the U.S., at least while the telephone company continues to provide the connection to your home, they are requried to allow other companies to actually be your ISP. Thus, while your phone company might offer you internet connectivity, you might also be able to look around and find additional DSL options from different, often local, ISPs.

For completeness, I should also mention fiberoptical connections, such as FiOS, which are provided by your telephone company, or satellite internet, provided by satellite TV or internet companies. The former has limited availability and the latter suffers from technical issues that make it only a last resort option when nothing else is available, in my opinion.
In your case, I'd at least start checking for the availability of DSL and how its pricing compares in your area.
A third, paid option
Mobile broadband is essentially wireless internet connectivity provided by mobile phone carriers and might be a possible alternative.
As you might expect, there are several options here because there are typicaly several mobile carriers that offer their services in any given region. Most offer devices that can be connected via USB or even standalone wireless routers that allow you to connect to the internet using the mobile network.
I'm not all that hopeful that they'll be as cost effective as a replacement for what you have but they're worth investigating. Be aware that there may be limitations on how much you can do (data transfer limits) and it'll almost certainly be slower than the cable connection that you have today.
Free options
Options for connecting to the internet for free are very, very limited. Essentially, you end up relying on the generosity of someone else who's paying for a connection to share it with you.

Your neighbor - Even though I advise against it for most, getting internet through your neighbor could be an option, depending on your relationship with your neighbor. If they have aWiFi access point and its signal is strong enough for you to connect to it from your home, it could work. All that you'd need is a WiFi adapter for your PC if it doesn't already have one. They'll have to trust you not to abuse their connection and stay secure, and you'll have to trust that they won't snoop on the data that you send through their connection.
The local library - Usually, this means using their equipment, but some provide free WiFi that you can use by bringing your WiFi-enabled laptop.
The local coffee shop or other generous business - For me, this means Starbucks, but many coffee shops and other small businesses are offering free WiFi as a perk for their customers. This does mean that you'll need to bring in your laptop, though.