If you don't want to bother setting up a server (and power to you, because server maintenance is annoying) just register with any open Matrix server you deem reliable enough. The matrix.org one is (obviously) pretty popular. As an end user, the federation stuff is no different from your average email address; there's a domain you store your stuff on and send your stuff through, that server is part of your address. If you make an account with a service provider that goes down, your messages disappear, same thing as would happen if Gmail or iCloud would take their servers down.
If you want the security of your domain but none of the hassle of managing a server, you can get managed Matrix servers from different providers these days [0]. Just get your own domain like normal, so your address will always be your own property and you can take it somewhere else if you really need to, and point the domain records at the servers of your provider.
If you do want to set up a server and join the Matrix network, there's an Ansible playbook [1] that'll set everything up on your server. You can also use the complete guide [2] if you want to manage everything manually. If you have any trouble getting federation to work, there's a nice diagnostic utility [3] that can help you identify the most common problems.
Alternative client are coming along nicely now, as well. For the longest time, encryption support was missing from the major ones (with "solutions" like running a pantalaimon instance in the mean time), but e2ee support has been added to most clients now. The only fully-featured client without encryption support I've come across has been GNOME's Fractal. On mobile, Fluffychat [4] has been working well for me, and on desktop Element [5] has been working well, too.
TL;DR: go to https://app.element.io/#/register, pick a username, and give it a try in your browser. You can join a bunch of the bridged IRC servers to get a feel of the conversation flow if you have no contacts on Matrix.
[0]: https://matrix.org/hosting/
[1]: https://github.com/spantaleev/matrix-docker-ansible-deploy
[2]: https://github.com/matrix-org/synapse/blob/master/INSTALL.md
[3]: https://federationtester.matrix.org/
The mobile app is a little unpolished. It constantly wants you to validate other sessions, always flashes the "loading" graphic whenever you switch to it, etc. Those two things feel a little half-baked--enough that my friends were like "no thanks" (I know, I know but, what can you do).
But broadly I still think it's very promising and would personally use it if I could get anyone else to. They're also building another server in Go that should have much better performance, if you're interested in that kind of thing.