Most of your early tames will be low level. The only high level tame you need to get early game is an Argentavis.
The best way to get prime early game is to trap and kill alpha raptors. If you're new to trapping alphas I suggest you place a stone 2x2 with only doorframes as walls with ramps from the top of the doorframes to the ground on one side. To trap the alpha raptor you simply get it to chase you up the ramp and fall into the trap. Once it's in the trap, you can shoot it with a crossbow.
The most important thing to do when using prime meat to tame is to let the Dino starve after you knock it out. This way they'll eat all the prime at once when you put it in their inventory.
Also, the prime meat will last 4x longer in a dinos inventory, so It's important to transfer all the prime into a Dino's inventory when transporting to the unconscious tame. Just make sure there is raw meat in your dino's inventory, or it may eat your prime.
A decent argent early on is a game changer. The argent will be able to kill alpha raptors for easy prime and you'll be able to pick up important dinos to tame. I wouldn't worry about getting high levels early on until you have the kibble.