Is there any privacy when surfing?
When you connect to the internet (legally) you will use an ISP (internet service provider). That ISP will have data on you in order to send you bills. When you start surfing the web all your requests go through this ISP and as such this company knows everything you do (i.e. They now that what websites you surf, what you download etc.)
How to improve you privacy from your ISP
First you could use a proxy server for surfing. If your country is blocking a certain website (for example because it hosts pirated software and movies) then changing the way you connect to the internet will circumvent this block. A nice list of proxy servers can be found here. There are several other good lists, a simple Google search can point you in the right direction.
The main advantage of these servers is that your personal data (for billing) is separated from your surfing behavior. Your ISP will only see you going to the proxy server website and although it is possible to log what you send to that website an ISP will not do that unless a court order is issued (simply because the amount of data to be stored to do this would be impractical). This is an important step in having a little more privacy.
Any website that allows you to use encryption of the communication channel (this means any website that starts with https) will stop the ISP from listening in to the communication. This is because the communication is secured (hence the s in https) by encryption. In practice this means your ISP can watch you navigate to your Gmail account but they cannot see your password since that is send through an encrypted channel, also all the things you do while in your Gmail are encrypted.
How to fool tracking through advertisements
On many websites you can see advertisements, many of these are hosted through one company. Whenever you visit a website any advertisement on that website can send your data back to the company that hosts the advertisement. The data captured could include your IP address, the details of your browser (and all plug ins installed), the referring website (i.e. Where you were before visiting this website) etc. If there is enough data such as installed plug ins and a tracking through cookies an advertisement company could create a pretty good profile on you.
So you want to block advertisements and tracking cookies. In Firefox there is on option to tell websites you do not want to be tracked. I have this enabled but I do not put a whole lot of confidence in this. I also block advertisements using a plugin (AdBlock plus).
As a second layer of security I also use the Ghostery plugin to make sure tracking is very difficult by blocking as many known trackers as possible.
I'm not claiming every advertisement company does this but there is no way for us to check this so better safe then sorry.
How to improve privacy by disabling statistics gathering on websites
There are software packages for webmasters that allow them to gather statistics on the usage of their website. Some of these packages have the ability to be shared with third parties or the software vendor. Google Analytics is a good example of this. You can share the data with third parties and you as a webmaster are supposed to make sure the data is anonymous. I'm not sure what the EULA says about sharing with Google itself, it is not explicitly mentioned that the data must then also be anonymised.
I'm not claiming that every webmaster misuses this data gathered during a visit on their website but again since we cannot check this I prefer not to give them my data.
I installed the NoScript plugin and that blocks a lot of these statistics gathering utilities. As an added benefit it also blocks any malicious script by default, I need to allow what scripts I want to run. This does change your surfing experience since any flash website will be blocked by default and you will need to manually (temporary!) allow this website in order to see the content.
What about Tor?
Tor is the next thing you might want to install in your browser. It will protect your data while on route on the internet (similar to having https). However the entry and exit points in the Tor network are not protected. Also it will slow down your surfing experience a lot. If you are not doing anything politically dangerous or illegal then the benefits of Tor do not outweigh the benefits in my opinion.
I do use the https everywhere plugin for surfing to ensure that for every site where there is a possibility to use https I actually do so.
What about surfing using another account?
There are many ways to achieve this.
The easiest is going to an internet cafe and surf from there. This might add a little privacy, the internet cafe has your personal data (often some form of ID is needed) and the only thing that changes is that it is now the internet cafe that has your personal data instead of the ISP. The internet cafe will probably keep your data and the logs of your internet usage for a while, probably there is a law somewhere that dictates what and how long they are supposed to keep this. It will be a lot more difficult for companies to get this information unless the internet cafe is actually selling this data (after removing personal information).
The same can be said for libraries. Also in libraries you will need some form of ID. Since libraries are often linked to the government it is actually even worse then surfing from an internet cafe, you might as well just send all your data immediately to Big Brother
Another way is to use another persons (wireless) internet access, this would add privacy but it is also illegal if done without permission.