You've got a few different things going on there. First, ~たりする is like "do X and other things." So 家で食事したりする is "eat (among doing other things) at home." Then のだ follows that, and it's negated -- のでない -- and that's made into its connective form: のでなく. I don't know how familiar you are with this grammar, so I don't know what to say about that really, but it's basically "don't just stay home and eat..." in terms of meaning. Or it could mean "I don't just stay home and eat..." depending on the context.
I'm pretty sure someone will be able to put up a link with explanations of both ~たり(する) and のだ, but I don't have any handy right now. Or maybe someone will have some good example sentences using them to help you out, but again, I'm lacking in that area.