Firstly, I take privacy online very seriously, especially when it comes to browsing and downloading stuff.
If you're half as paranoid as me, then you might want to seriously consider what I describe on this page.

As always, make sure that websites you visit are encrypted with SSL/TLS. I trust you have heard of one or both of these, if not, please research those terms and how they work before you read this article.

One of the best (bundled) solutions so far is to use services like the "Tor" network.
It allows the user to use the internet almost completely anonymously. You can find information on how it works on their website.
For tl;dr folks, I'll give a very brief, semi-technical description.

Whenever you want to go to a website, the Tor network will go through a couple of different servers (referred to as "nodes") before routing the traffic to the destination server.
yo!
In the above example we can see that first our traffic is routed through Tor's obfs4 obfuscator PT (Pluggable Transport) to prevent DPI (Deep Packet Inspection). More info on that can be found here and here.
Then the traffic is routed through two nodes located in France before finally reaching its destination labeled as "Internet".

Tor is designed to pick a new route every time a new website is visited. Originally, it was meant to be random, however, as its popularity increased, the network shifted towards bandwidth-based load balancing, therefore making routes somewhat predictable.

Tor Project is very privacy-conscious, so their Firefox-based browser (nicknamed "Torzilla") comes with a couple of plugins that help protect the user against spying and tracking.
However, as with most things, Tor isn't unbreakable. It's possible to "break" the user's anonimity if a single entity controls enough "exit" nodes (the last node in the chain which directs traffic to its destination). FBI claims to have done it (article).

Tor Project officialy supports connections through their browser only, however, it's possible to configure devices to use Tor as a proxy for all connections. As I mentioned at the beginning, this would enable the user to download content semi-anonymously, although this would only work if the user was downloading stuff directly from the server (using HTTP).
It's possible to use P2P (Peer-to-Peer) technology such as Bittorrent over Tor, but this creates severe security and privacy implications that defy the entire purpose of the network as detailed here, additionally, it would be VERY slow, and would slow down the network as a whole.

So, we're done with Tor, let's move on to DNS (Domain Name System).
At it's very base (and similarly to ARP (Address Resolution Protocol), DNS tells the requesting computer where another computer is located, however, this is done through the use of domains (e.g.: www.google.com or www.davidfahlstrom.se, they're both domains).
The problem with DNS, is that whoever controls the DNS servers used by devices can see what websites the user is visiting. By default, most devices use DNS servers provided by their ISP, but that means that despite not knowing what the user is doing on the website, they still know what website it is.
Luckily, there is a very easy solution to this, we can use third-party DNS servers. A German company IAMONSYS provides free IPv4 and IPv6 DNS servers which they claim are completely unfiltered and unlogged. Personally, I've been using them for a while, and have no complaints.
hey!
Above image is of DNS configuration on my computer, and these are the primary and secondary DNS servers they provide, more information can be found on their website at dns.watch.

Lastly, another solution (in addition to what I mentioned above) would be VPN (Virtual Private Network), it works by allowing the client (user) device to connect to the device and become part of the network. By connecting to the network, the device can also access any resources available locally (permission-dependent), it also assumes the VPN server's public IP (Internet Protocol) address.
This is a rather secure solution when configured and used properly.

This article was written in 2 hours, if you feel like something should be added or corrected, please feel free to contact me at: contact AT davidfahlstrom DOT se