How do I hunt most effectively?

I am at the stage where Lara becomes able to set up camps and hunt animals with bow and arrow to start with.

However, I seem to not hit anything: Deers run away mostly, not to mention the swifter rabbits.

So, the main question is, do I need to learn this stuff to advance in the game, or is it a side gimmick? Can I get more powerful weapons somewhere along the story line, or do I need to craft that stuff?

Do I need to learn this stuff to advance in the game, or is it a side gimmick?

Overall, hunting is a side gimmick and has no bearing on the overall progression of the story. However, hunting is required to gain materials to upgrade your equipment later on.

If you want to hunt effectively, build up your hunter tree. At the beginning of the game, hunting is a little tricky due to the lack of better gear and skills. Some skills, like Animal Instincts and Anatomical Knowledge, can be obtained early and will allow you to see/aim at the animal easier while using Survival Instinct.

Can I get more powerful weapons somewhere along the story line, or do I need to craft that stuff?

As you progress in the game, you will obtain more powerful gear that will make hunting easier. Some weapons are obtained through the story and others require you to locate strongboxes to find weapon parts.

I can't remember exactly when, but you can eventually craft upgrades for your weapons to provide improved accuracy, damage, rate of fire, and more.