The problem#
Signal is a privacy-focused instant messaging application which ensures end-to-end encryption on all messages and generally goes out of its way to avoid accidentally revealing your conversations to third parties. As is common with security mechanisms, this necessarily adds some friction some some tasks you actually want to do. In particular, this means viewing your message history outside of the official app is unsupported. Which is especially a problem due to the app being tied to a smartphone that will likely only continue being usable for at most several years. So there's no official way to maintain access to your conversations indefinitely in contrast to IRC and Pidgin logs I have going back decades.
The solution#
The Signal Andriod app supports making encrypted backups (there is no way to get messages out of Signal iOS).
When you enable backups, it gives you a "passphrase" (a sequence of 30 digits). If you have lost this passphrase, there is no way to recover it, but you can disable backups and then re-enable them which will generate a fresh passphrase that will be used for your backups going forward. I recommend storing this passphrase in a password manager like KeePassXC.
To actually use the backup file, you'll need to know where it is. The location is shown on the Signal settings under Chats -> Chat backups -> Backup folder. You can copy it to your computer by plugging the phone in with a USB cable and using an MTP client, which is included in most file managers (I use XFCE's Thunar).
Once you have the backup file and passphrase, you can use
signalbackup-tools to convert the backup to
something human-readable:
signalbackup-tools [input] [passphrase] \
--exporthtml [directory] --allhtmlpages
The output does a good job of trying to look like the Signal interface. See the documentation for more options on controlling the output including various options to filter the output to just some conversations or time ranges.