There are some automated messages sent when people first sign up that you can customise to include any relevant info you think would be useful, as well as signpost useful places to start, or encourage them to introduce themselves in a dedicated category, for instance. These can also be sent from a specific user, rather than @system, which gives them a little more character. There’s a bit of info in Welcome new users with customized trust level promotion messages that you may find useful. There’s also the Automation plugin which has a ‘send pms’ script you can use to set up any extra ones too (with a useful delay so you can pace the messages).
There’s no way for regular users to look up new TL0s easily, but that trust level is meant to be a very temporary one and people will progress to TL1 after a short time of reading:
But to see a list of new TL1s, anyone can check out the Basic badge list (if it’s enabled) and the latest people who have been awarded it should be at the top - Discourse Meta
You may want to consider what a sitewide ‘welcome’ etiquette would be for your site as well. It could be quite overwhelming for some to receive multiple personal messages from different people when first arriving/not having posted yet. Some people may need a little time to acclimatise and settle in first.
You can use the data explorer to create custom queries (which can be made available to certain groups so they can pop them in their sidebar). It could be a simple list of new signups, but you could also use these to generate a list of new people who you think meet specific criteria and would be suitable to reach out to (in a coordinated effort with your TL3s).