The EvaCoop-Team forked eosjs to make it more compatible with react-native. Async handling in react-native is a big mess with the original eosjs.
Here's the link to the git react-native-eos which also contains examples.
Probably the best tutorial is to take a look at the Universal Authenticator Library - UAL and how it works.
Here are some helpful links:
Medium Article about the UAL
UAL Authenticator Walkthrough
There are are several different UAL-Repositories + Usage-Examples for use with the different available Wallets.
If you just want to read data you can use the EOS RPC API.
Then you can use the Fetch API (for React Native) to connect to it promised based.
Your code could look something like this to read from a table of a contract.
How does one attach a message using a desktop wallet or cleos?
What is the max limit of the message?
How much does it cost to send a 1000 character message?
Answered here if you don't want to send EOS with your transaction, and here if sending EOS is acceptable.
What stops me from spamming the same message to every account and bloating the blockchain?