I have various Kindles including:
My Kindle Fire first model, second edition, WiFi - it can access any web site over wifi
My Kindle 4 WiFi and Kindle 5 WiFi can also access any site over wifi
My Kindle Paperwhite 3G can access Wikipedia.com over 3G. It asks for a WiFi connection to visit sites such as google.com This concurs with Nathan Osman's comment on your question.
So in answer to your question, all the kindle models I have include a web browser. They can access most sites using WiFi, however the newer 3G models are restricted to wikipedia and amazon by Amazon.
PS: Even on the older Kindle Keyboard 3G models I found they sometimes asked for WiFi when I tried to visit sites after a while. However if I first visited the Amazon store then the 3G connection worked correctly for other web sites. I tried this approach on the Paperwhite 3G however I was still asked for a WiFi connection when visiting sites such as Google so there seems to be a policy-based restriction on newer models (possibly to save Amazon from data charges?)