Google search is notably dangerous for privacy. The reason that Google can provide it's search services for free is by tracking your search history and proving that they can target their ads to you better than any other service. Bing, Yahoo, and most other search providers are similar.
Better options include:
DuckDuckGo: The closest drop-in replacement for a search engine like Google. Firefox has built-in support for DuckDuckGo searching.
Searx: A federated metasearch engine that searches multiple engines and pulls the information while removing all the trackers. Very useful for deeper research projects that an individual search engine might not find everything for. Many instances are available, including a Tor Hidden Service.
Browser plugins are one easiest of the ways you can help secure your internet browsing experience. Generally, you'll want:
An Ad Blocker. This not only stops annoying ads, but also blocks a lot of trackers that allow companies to spy on your browsing history. We recommend uBlock Origin, but personal preference can be a factor in choosing.
HTTPS Everywhere. This plugin is a collaboration between the tor project and the electronic frontier foundation, two of the biggest names in privacy. This plugin makes sure that any website that supports the encrypted HTTPS standard uses it, making the content on the sites you visit invisible to your ISP and anyone else with access to your network.
Decentraleyes. This plugin blocks requests from big ad-serving services like Google and Amazon and replaces it with locally hosted content to ensure that your information is kept where it belongs.
Firefox is generally the browser of choice when thinking about privacy concerns. If you are uncomfortable with Mozilla having any user data, turning off their analytics program will guarantee that they don't collect any information on your browsing habits. All of the plugins above work with Firefox and Firefox-like browsers.
Google Chrome is very much a privacy nightmare. Google analytics is very good at collecting information about it's users, and you should generally not trust any Google products. Chromium is better, but still not ideal. For a Google-less browsing experience with a chrome-like UI and features, ungoogled-chromium is a good choice. Alternatively, check out Brave, which uses a lot of code from Chrome.
Safari, Internet Explorer, and Microsoft Edge should all be considered entirely insecure. Not only does Edge record usage data, but the fact that these browsers are often used by the people most likely to be compromised means that a large amount of time has been spent finding their vulnerabilities. Avoid these browsers.
DO NOT use any browser that comes with a VPN (e.g. Opera). These may hide you from an employer, your ISP, or other people on your network, but if you're not paying for it, they're just logging your data to serve you ads better. If you can't afford to pay for a VPN, or don't trust any VPN, use the Tor browser.
The Tor Browser the simplest way to access the Tor Network, which is a feat of computer science and a massive community undertaking. While the entirety of the network is too complex to explain here, it basically functions by encrypting your internet traffic, sending it to an entrance node, and routing it through several servers, each stripping off one layer of encryption, until it reaches an exit node, where it is sent to the server you actually wanted to contact. This way, no server has access to both the sender or receiver of the information and the actual content that it's sending or receiving. It's generally trusted as the best way to anonymize your network traffic.
You can download it here. Make sure to verify the signature using the guide above in general tips.
Tor is not perfect, however. At the exit node, the last of the Tor encryption is stripped off, so anything unencrypted can be read by the exit node or any servers it passes through before returning to Tor. Making sure that your traffic is encrypted by other means, specifically using websites that support https:// instead of http://, add another layer of protection. A more complete understanding of vulnerabilities can be found here.
A VPN, or Virtual Private Network, is basically like an encrypted tunnel between you and the rest of the internet so your ISP, employer, pesky government agency, or any other potential invader can't see what you're doing and the sites you visit can't see who you are. VPN's can range from very cheap to pretty expensive, depending on your needs. A VPN is absolutely recommended whenever possible. If you can't afford or justify a VPN, Tor is a very competent replacement privacy-wise, but will often be significantly slower.
DO NOT use browsers with VPN's built in, for reasons above, in the browser section.
Theoretically, a person or organization can't have their equipment searched or tampered with by most governments without a proper warrant. While this idea has gotten thrown out the window occasionally, it still generally applies. However, most of the time, a group cannot say that a warrant has been issued against them. To circumvent this, many websites that are in danger of being exploited by government agencies to tap into user's privacy issue warrant canaries, which merely say that a warrant or gag order has not been issued against them. You can find raddle's warrant canary here. If you're worried about a website being compromised, check for a canary. If it has one, and it's updated regularly, you're probably "safe".