재미있는 이야기도 많이 하고 아주 즐거웠습니다.
Is this a full sentence? I understand it as "many interesting stories and enjoyed them." So although the subject is omitted but it seems to be a full sentence.
Also, what is the meaning of 도 in 이야기도?
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Yes, it is a full sentence - from a native Korean's perspective. Based on the meaning of the sentence you posted, one can presume that there was some kind of meeting right before the conversation. So it is okay to assume that the subject(some kind of meeting) is omitted.
And you can understand 도 as 'and': another casual/informal omission, like '재미있는 이야기도 많이 하고 (맛있는 것도 많이 먹고)'.