How can I make payments on the Internet without leaving a money trail?

To protect your privacy and avoid tracking, nothing beats cash.

There are various services that let you purchase credits in cash at a brick and mortar store, which you can then later use to purchase goods and services online.

One example is paysafecard (I haven't tried it, but you should also be able to buy bitcoin with cash). There are a number of VPN providers which accept these payments.

Alternatively, you could simply purchase your VPN access directly offline at a store.

There are also domain registrars which accept these payment methods, but most will ask for identifying information (name, address, etc) when registering a domain. So if you want to conform with registrars TOS, registering a domain anonymously wouldn't be possible. You can hide your information from third parties by requesting that your registrar doesn't disclose the information, violate the TOS by providing false information (not recommended), or find a registrar or third party service that does not request this information.

Using a VPN won't make you anonymous.

The owner of the VPN service will still knows who you are from the origin IP address.

In the past several VPN providers that stated that everything was anonymous in fact released all the information they had to the FBI and US Department of Justice. It makes sense that they are not private; if you commit a crime using their systems and they cannot point to the perpetrator of the crime then they might be held liable.

For the crypto currencies, you do not need to buy them; you can mine them. Also some crypto-currencies have a higher level of privacy than others.

At the moment the best possible way to make you private in some of the internet traffic is still Tor as long you follow their best practice.

The other way to be private in the internet is if you steal someonelse's digital identity. This is the most common means used by thieves.

You could use xmr.to for that. According to them:

XMR.TO allows you to make a Bitcoin payment with the strong privacy provided by Monero. This means that using Monero together with XMR.TO, you can pay any Bitcoin address in the world while protecting your privacy.

You buy Monero, a privacy-centric coin, send to xmr.to with the destination Bitcoin address, they sell your Monero, buy Bitcoin, and send to the host. Xmr.to a well known service, and lots of people uses it every day.

Their FAQ is pretty informative, and you should read it.