I am trying to share contacts through text messages just as I used to do with my iphone 3 but I do not know the correct taps and/or gestures to do this.
The easiest way for sending contact to another person is through the Contacts App. Just tap on the contact you want to share and you'll see the details for that person. At the bottom of the page is a button "Share Contact". There you can either choose to send this through a Message or email.