I'm not sure how the Iphone works, I have an android phone and I'd assume they work in a similar way.
Here's how android works:
Open up the conversation with the selected person.
From here, if you press and hold on a selected message, it will give you the option to delete that particular message. You can delete individual messages this way.
If you wish to delete more messages, in the conversation there's a tab in the top right corner, where you can click delete and then you can select messages by ticking the boxes on the ones you want to delete.
Like I said though, this is for android phones - but hopefully the IOS will work in a similar way :)