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 interface, 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
A common source of internet connectivity failure is your OpenWrt and upstream WAN routers having conflicting IP address ranges. To correct this, adjust the OpenWrt router subnet by changing the network settings in the file /etc/config/network
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 OpenWrt is configured as a router with default settings, you can access LuCI web interface and SSH command-line interface by domain name. If OpenWrt is configured as a network client, switch or access point, use its IP-address explicitly. You can utilize a network scanner or status page of your main router to find out OpenWrt IP-address.
If you have flashed a development/snapshot firmware of OpenWrt, you first need to manually enable the web interface: development installation guide. Or verify the result on snapshot builds connecting to OpenWrt via SSH.
You can consult the troubleshooting section of the User Guide, if you think that resetting OpenWrt's settings might help.