Go to a blockchain explorer site such as blockchain.info, BlockCypher, or Block Explorer in a new browser window. Sites such as these can be used to view all bitcoin payments made on the blockchain. There are many sites that can also search the blockchain, and it's up to personal preference which one you use.
In the 'Search' area, you'll need to enter information that is specific to your bitcoin transaction. Typically, this would be the bitcoin address you made the payment to.
Once you select 'search', all bitcoin transactions that involve that address will be displayed, newest to oldest.
If there are a large number of transactions involving that bitcoin address, the easiest way to navigate to your particular transaction is to use your browser's Find feature to search for the exact bitcoin amount you sent. This is usually under 'Edit > Find', or you can use the keyboard shortcut Command + F (macOS) or Control + F (Windows).
Having found the correct transaction, the transaction ID is displayed as a long string of letters and numbers in the grey bar above the amount from the previous step, just to the left of the transaction time.
To view the full details of the bitcoin payment, you can visit the transaction ID link. You can then share this link - or just the transaction ID - with your recipient.