Reminisce doesn't have a "happy remembering" meaning, nor does it imply spending any time with anyone, friends or family or otherwise. Reminisce only means "to remember, talk, or write about the past (in an indulging way)". Thinking about "happy things in the past" as the question wanted is more like "to reminisce fondly".
The closest match in Chinese is
追憶 / 追忆, meaning "to remember past events". It is similar to
追憶 evokes a stronger, more emphatic feel, making it more similar to English's "to reminisce".
懷舊 / 怀旧 conveys a sense of cherishing the memories. Its means "yearning for the past", rather than recalling it, and implies rose-tinted glasses. Note that
懷舊 is also a noun, meaning "nostalgia".
Another suggestion that can't be used for reminisce is
懷緬, or more correctly
緬懷. It actually means "imagining the past (history, not memory)" or "cherishing the memory (typically, of someone who died)". For example, "Commemorating my late mother (
緬懷天上的媽媽)". Rather inappropriate for translating reminisce.
What the question really asks for seems to be "to reminisce with friends and family". For this sense of the word, a good Chinese option is
敘舊 / 叙旧 - literally, "to talk about olden days". This word is strictly about reminiscing with people, and works well for situations such as "to reminisce with old friends at the high school reunion (
在高中同學會與老友敘舊)". Note that
敘舊 can't be used for general thinking, remembering, or writing.