The very first thing you should do is to use your brain and ask yourself why you should not choose free services in most cases.
If an Internet service costs nothing, how does it finance itself? Mostly with the sale of data, your data. There are other ways, see TOR, but this must be transparent and established.
Second thing, to move anonymously on the internet, you need an access that is not connected to your real identity. Two (three) possibilities, use a foreign access, is usually not simply available, but it is a possibility. Second possibility, choose a public access, without having to give your data when using it.
Or last possibility to mask the IP registered in your name by services like TOR and/or VPN/Proxies.
An Internet access that is reported to a fake identity would also be possible.
These are the basic requirements and that all paid services (except the registered Internet access) were paid with an anonymous payment option and no real registration data (including the used IP) was used. Thus, the entry usually starts with the TOR browser.
That is the basic framework. The basic requirement and when using such methods it is to avoid 100% to login, to link or even to mention your real data, also e.g. usernames from the clearnet which are connected to you. For this reason, the extra device is also often used for this, a secure laptop or the LiveOS Talis on VM or USB, to avoid the mixing.
Every communication/connection should be encrypted ssl/tls/openvpn/pgp to name the most important ones.
Then look around, there are enough providers and compare the offer, choose carefully what suits you and try to avoid the free providers.
For oneway mails, or simple registrations you can use disposable addresses and free services under masking your IP, but for a fixed address that you set up as an anonymous contact for yourself you would be well advised to look for a decent mail provider.