The /b/ sound is created by bringing both lips together and releasing a burst of air. The /ɛ/ sound is produced by opening the mouth slightly and lowering the jaw while keeping the lips relaxed. The /r/ sound is made by curling the tip of the tongue slightly upwards towards the alveolar ridge, without the tongue touching the roof of the mouth.
So, in summary, the phonetic transcription of "bear" is /bɛr/, with the initial sound being a voiced bilabial plosive /b/, followed by an open-mid front unrounded vowel /ɛ/, and ending with an alveolar approximant /r/.