There are many ways to use the internet for free, and it really boils down to whether you have your own computer/smartphone or not.
If you don't have a computer or smartphone
The best way to access the internet in this case is by using the many free computer terminals available in your community. Often the best answer is to use the computers at your local library. There may be set times or a sign-up sheet, but most communities libraries offer free computer use.
You can also check and see if there are any free computers to use at a local college, adult center or senior center.
If you need to, you can also access the internet at electronics superstores like Best Buy and Frys. Just use the demo computers for any quick needs (don't be on those all day!). Apple stores also give you access to their test computers and the internet.
Hotels often have a business center with computer stations. Use those when traveling. Hostels also usually have a computer for the use of their patrons.
Travelers can often find free computer kiosks in airports, at Student Union buildings at universities and often in local internet cafes around the world.
Another tip - my town's visitor center/chamber of commerce building has a long desk with several computers set up for free internet use by the community. I could go there every weekday if I needed this option. See if you have something like this in your town.
If you have a laptop
If you have your own laptop, all you need is a working WIFI card (usually this is already installed inside your laptop). There are so many places around the world with free WIFI hot spots - entire websites are set up just to help people in every country, state and city find free internet access. Places to access free internet include:
If you have a smartphone
Anywhere your phone carrier has coverage, you can be online for free with a smartphone, surfing the net at your leisure. These connections can be clunky and your options for doing much 'work' might be limited, but you still have the power of the net in your hands.
Tablets are even better smartphones for anywhere access, if you are set up with a phone company.
If you have a Kindle device you will also have free access to the internet. The connections are even clunker than using a smartphone, but it might be better than nothing - you could certainly check your email. :)