Here's some specific popular dinosaurs and where you can unlock them - including all the ones that are only available through Contract missions:
Building an Expedition Center on every island unlocks new dig sites for all islands, so while you can't unlock every dinosaur from one island you can go back to an earlier, more profitable island to start sending expedition teams out - and then put those dinosaurs on any island. Once you hit 50% Genome you've unlocked that dinosaur, although you'll need a much higher genome if you want to start splicing or you don't want the dinosaur to die during incubation. Click the dig you want to go to - it'll have all the dinosaur fossils that can be dug up listed next to it - and wait for the fossils to come back, then analyse the good ones and sell the useless ones. Build an Expedition Center and a Fossil Center and you can send teams on digs around the world to uncover and ultimately unlock new dinosaur specimens.
You don't get all dinosaurs at once, they have to be discovered as you go through dig sites.
