Deploy to devnet, then mainnet
Devnet first
Set the cluster in Anchor.toml (or pass the provider flag), fund the deploy wallet with a devnet airdrop, and deploy.
solana config set --url devnet
anchor deploy --provider.cluster devnet
solana program show <PROGRAM_ID>Exercise it like a user
Run your client against devnet, check transactions in an explorer, and confirm upgrade authority and account sizes are what you expect.
Mainnet
Use a dedicated, well-funded deployer key, and consider a multisig (such as Squads) as upgrade authority. Publish a verifiable build so users can match on-chain bytecode to source.
Priority fees and reliability
Mainnet transactions compete for block space, so clients should set compute-unit limits and prices.
Commands and APIs change, so the official Solana docs and Anchor docs are the source of truth.