When you start doing tai chi, you should breathe naturally, unless you are told otherwise by your instructor.

In tai chi form learning, your breathing will eventually settle down, as you keep practising the form. While some schools teach specific breathing for each sequence, if you just keep doing tai chi, then your breathing will come right of itself. Relax and breathe normally and you will be fine.

Being told otherwise by your instructor will happen, if you are learning chi kung (qigong), as these are specific movements, designed to help deepen your breathing and circulate your energy. Most people breath with shallow breaths, meaning the air at the bottom of the lungs can get stale. Deep breathing will clean out this stale air. Chi Kung also addresses your energy - if you come to it feeling exhausted, you will feel much more alive and energised afterwards.

There are many, many different chi kung sets out there - don't even try to learn it from a book or video for 4 reasons:
Firstly, some of the stuff in those books is aimed at advanced students, and beginner students will suffer adverse effects, since their bodies are unprepared. Like diving before you've learned the doggy paddle...
Secondly, your instructor can correct your movements. Even if you're just copying, it's very easy to miss observing something critical, like weight shifts, position of feet, knees, etc. Your instructor will gently help you correct these, so that you get the most benefit from the set.
Thirdly, many, if not most, schools only teach chi kung from the teacher to the student, it's never written down or videod. This means what's in those books / videos that are out there may not even be fully correct.
Lastly, some schools will teach progressively different chi kung sets, as you progress up through the levels - as your body becomes more loose and relaxed and able to do more. Using a book or video means you won't know what the differences are and won't be able to progress upwards in your own practise of chi kung.

Anyway, relax and keep breathing!