быть = to be

бывать = to be [repeatedly], to be present; to visit

Examples:

Вчера я был дома = I was at home yesterday [for a defined, short period ]

В марте я бывал дома каждый вечер = I was at home every evening in March [I was at home on repeated occasions]

Я был во Франции = I was in France - a simple confirmation of the fact that I was in France [on a particular occasion]

Я бывал во Франции = I visited France - means I was in France and on more than one occasion.

бывалый ( adj. same root as бывать) = worldly-wise / one who has seen and experienced much in life.

also = ex - [now usually бывший]

бывший муж = ex-husband / former husband

бывалый солдат = veteran [more usual now - ветеран]

Это бывалое дело = that's nothing new / that's normal, everyday