To get an egg, you will need to explore the Mysterious Cave.
Feed the T-Rex 80 Food and it will give you 50 Gold and patrol for 60:00.
If you start a deal at the beginning of those 60:00, you lose them. They do continue after the time is up on the Quick deal, but they do not add onto the end of the Standard, Lengthy, or Long-Term deals. To get the best use of your 80 food, wait till the 60:00 is almost up and then start a deal.
If the T-Rex is patrolling alone, when you just feed him and don't start a deal, he will walk around the village. He will only run when a Murlod gets within a certain proximity of the T-rex.
If the dino is off somewhere elsewhere on the map, the timer on the deal does not start until the dino gets back to the lair and picks up his rider. Switching to another island or going to the Zepplin map starts the timer instantly.
If you start a deal and have a rider on the T-Rex, he will run for the entire duration of the deal.
With the above in mind, the best food saving method is feed the dino, wait 55 minutes then do a long term deal, this gets you an active dino for just under 4 hours per feed.