I have found this sentence: "Kazoku de dekakeru". Why is the particle で used instead of と?
The difference between using で and と is the difference between "we went out as a family" and "I went out with my family", I think of it as rather like the French "en famille". 家族で is very common expression and you can also hear this grammatical use when people refer to doing things as a group (グループで／皆でやりましょう).
It is another variation on the use of the で particle to indicate the means/use of something to do something else (映画をテレビで見た、日本語で話した etc) although I must admit I find it easier to remember and understand intuitively rather than by trying to rationalise this definition.
(The easiest way I think of to explain the double use で is to compare it to the use of "how" in English: The question "How did you go to the races?" can grammatically be answered in two ways: "As a group/as a family" or "By car". The difference is that where as in English we use the same word to make two different inquiries, in Japanese we use the same particle to give two different answers to [what would be] the same inquiry [in English].)