Matrix is good for private general messaging. The fact that it's decentralised means it can also withstand things like government-ordered shutdowns or back doors, since there is no central point that controls the whole network.

Two things to be aware of:

• Some non-message bits (e.g. room topic text and membership) have not yet been moved to the encrypted channel, so those could be read by the administrator of a homeserver that participates in your chat room. Since most people care primarily about keeping the message content private, this is an acceptable trade-off to get all the things that Matrix offers.
• The upcoming Matrix 2.0 features and design choices simplify the UI and fix some occasional errors. It might be worth waiting until this stuff officially lands in the client apps before bringing your contacts to Matrix, for a better experience all around.