For a first quick Internet access test: If you have an existing router, connect the WAN port of your OpenWrt device to a LAN port of that router and confirm internet connectivity of your OpenWrt device with the following steps:
In the OpenWrt admin web GUI, go to Network → Diagnostics and Click on “ping” button
or, if using OpenWrt SSH command line, you can use the command ping openwrt.org
This should return “0% packet loss” if everything is alright with your Internet connection.
When using your OpenWrt device as a Wi-Fi access point, please remember to initially set your country code in the OpenWrt Wi-Fi configuration, to properly comply with your country's Wi-Fi legal regulations!, e.g. see here for a first basic Wi-Fi setup.
Consult the User Guide for more advanced configuration.
Install custom software packages that you might be interested in.
Troubleshooting your first steps with the new OpenWrt device:
If you have flashed a development/snapshot firmware of OpenWrt, you first need to manually enable the admin web GUI: development installation guide. Or verify the result on snapshot builds by SSH-connecting to your OpenWrt device IP 192.168.1.1
You can consult the troubleshooting section of the User Guide, if you think that resetting OpenWrt's settings might help.