I have read on multiple websites about an Anti-detect browser used to anonymize web surfing better. Can I achieve a similar level of anonymity without installing the software and just changing system and browser settings myself? I just don't trust those crack downloads of the software and it's just too expensive. Does using a VPN service leak browser info?
There is the option to send a Do Not Track request with your browsing traffic in Chrome. Go to Settings -> Advanced -> Privacy and Security.There is something similar in Firefox.
There are also a couple of (completely safe) Chrome extensions from the Electronic Frontier Foundation that will help you: Privacy Badger blocks trackers such as certain ads and cookies, while HTTPS everywhere makes sure websites use HTTPS instead of unencrypted HTTP.
Unfortunately, these extensions won't work on a phone. The Electronic Frontier Foundation recommends using the Warp browser instead. This is a Chrome-based browser that looks a lot like Chrome. I have scanned it with Bitdefender and it did not produce any malicious alerts. It has HTTPS Everywhere built into it, as well as Privacy Badger, and no ads. It's also very fast, and available on the Google Play store (make sure to go to the genuine page: follow the link from the EFF website).
All of these are completely free and very easy to use/install.
Using a VPN, as long as it is reputable, can improve privacy but many are either not free or have a data limit.
Using incognito mode means your browser does not save history or cookies, but websites will still know you visited them, and your internet service provider can still track what you are doing.
There are many purpose built browsers or forks of major browsers that offer security addons. However, with some searching you can find guides that will allow you to manually change settings to improve security.
There are also plenty of other tools and addons you can install like application sandboxes and add blockers. Many of these are free and will have reviews and discussion on the web.
Some of the more privacy focused tools like VPNs or using TOR will require you to use additional software, but you should be able to research and find credible reviews for anything you want to use. Security.SE is not a good place for product recommendations, so you may want to try other forums to discuss specific tools/products.
I think you don't need any special software to stay completely anonymous on the internet. If you got the Tor browser you could browse normally anonymously. Tor is made to make people completely anonymous on the internet by encrypting your data and connecting you to the Tor network. It directs your traffic to servers in the Tor network that encrypts your data:
From the offical Tor site:
"Tor helps to reduce the risks of both simple and sophisticated traffic analysis by distributing your transactions over several places on the Internet, so no single point can link you to your destination. The idea is similar to using a twisty, hard-to-follow route in order to throw off somebody who is tailing you — and then periodically erasing your footprints. Instead of taking a direct route from source to destination, data packets on the Tor network take a random pathway through several relays that cover your tracks so no observer at any single point can tell where the data came from or where it's going."
So you may have to change a browser but you will be really safe with Tor as it's actually covering your tracks like an attacker would do to get away with a crime. Also, a small bonus...it's possible to view so called onion links that is a one encrypted link that's only possible to run on Tor and hard to find unless somebody gave it to you or you are using Tor.
Tor is really safe so if you just download it from the official website and keep it updated you should be fine. Here is their website: https://www.torproject.org/