The problem is heuristics. All mentioned tools are built on heuristics and the only way to avoid them is to change how you live completely. You can be fingerprinted by the modules installed in your browser. By the programs you use and the frequency you use them.
These days you’re going further than just online behavior. Shops know what you buy in what amounts, because nobody buys all the same brands you are getting fingerprinted constantly. This is used for targeted advertising, but it can also theoretically be used to track you.
MIT’s Reality Mining project proved the same using smartphones. You prefer certain apps, you use your phone at certain intervals, you move around certain places. This all contributes to a somewhat unique pattern (back when I did some research on it during my internship we were getting 91% certainty in simulations, even when people changed their SIM card every few days we were still able to track them based on the SSIDs they encountered, places they went, apps they installed and used, when they checked their phones, Bluetooth devices they connect to, cell towers they passed at a certain moment in time and what smileys they use in text messages).
Avoiding heuristics means changing everything you do completely. Stop using the same apps, accounts, go live somewhere else and do not buy the same food from the same brands. The problem here is that this might also pop up as a special pattern because it is so atypical.
Changing your identity is the first step. The second one is not being discovered. As Thomas said the internet doesn’t forget. This means that photos of you will remain online, messages you posted, maybe even IDs you shared will remain on the net. So even when changing your behavior it only will need one picture which might expose you.
You cannot enforce forgetfulness. The Web is like a big memory, and you cannot force it to forget everything about you(*). The only way, thus, is to change your identity so that everything the Web knows about you becomes stale. From a cryptographic point of view, this is the same case as with a secret value shared by members of a group: to evict a group member, you have to change the secret value.
(*) Except by applying sufficiently excessive force. A global thermonuclear war, with all the involved EMP, might do the trick, albeit with some side effects.
I agree with one of the comments that one could write a book on this (I have an idea!) because this could get broad, but consider that these data mining tools (and I build them!) make some major assumptions. For instance, consider search history. One could easily build a program or a tool that would do “random” searches on topics where a person lacks interest. I’ve done this with Google where a program will search for dogs, dog food or dog treats and months later, I suddenly see these advertisements appearing everywhere.
if we conduct any transactions online these can all be collated to
build a good picture of what we do, buy, read etc
When it comes to finances, it’s similar; I have to make an assumption that the data I receive are an accurate indicator of who you are. Suppose you make 1/3 or more of your purchases completely away from your interest, for instance, you’re truly a Libertarian, but you decide to subscribe to a Socialist magazine. How accurate are my data then? Also, you may change in ten years, so how accurate will my data be then, unless I account for it (and how effective then is it to have all the historic data)?
Of course, we could all argue, who in the world would do that? If you’ve ever read Soros’ The New Paradigm For Financial Markets he mentions his family went through WWII without the trouble many Jews had because, as I interpret it, his family was careful with the information they provided to others. Privacy, historically, is priceless, even if a few generations mistakenly think it’s not.
As a note, this answer only addresses the above concern with money.