First, make sure you really want to serve in this way--
it is an opportunity for service, but don't do it out of obligation. (Hopefully, you do feel motivated to serve in this way!)
Second, ask the person in question!
Call, email, text, ask in person.... it doesn't really matter. What you say is more important than the medium you choose. For instance, don't say, "Do you need anything? I'd love to help you out." The person will likely say, "Oh, thanks, but I think we're okay."
Instead, be very specific. Say something like this: "I know times are rough for you guys right now. I'd really love to bring you a meal to help you out. What day is good for you?" OR "I'm going to the store tomorrow and wondered if I can pick anything up for you. I can swing by your place on my way home. Do you need any milk? or cereal? Want me to pick up a rotisserie chicken?" OR "I know you're moving Friday, and I'm sure you're quite busy this week! Please let me bring something by tomorrow for you and your family to eat for dinner. What do your kids like to eat?"
You get the picture. A specific request/offer like those above shows the person in question that you really are serious and want to do this. Sometimes, you can assume that they will accept (for instance, when a mom has just had a baby!), and just announce you're coming by in a day or two.
If your church has people in place already who coordinate these types of ventures, seek that person out and volunteer. Our nursery coordinator takes care of new mothers, our women's ministry directors help with meals related to a death, and we have a woman who coordinates a freezer in the church kitchen, making sure it's stocked with various things someone could to take to a widow or shut-in on a visitation. The other occasions (morning sickness, moving at the end of the week, child in the hospital, and so forth) are just done by whoever sees the need. There are a few of us who regularly do this sort of thing, and we usually send an email around asking if anyone's gotten the ball rolling yet. If there is someone official who takes care of meals, you are better off signing up on their rotation or volunteering to be on their list. Otherwise, you might take more food than can be eaten by the family in question or "step on someone's toes" if this is their "official ministry." I can guarantee that those who normally coordinate meals love to have a "regular" they can count on when a need arises!
Once you've figured out a need and/or volunteered to be on a coordinator's list, what do you take to the family in question? What if you don't cook? What if you work all day and don't have time to fix a "home-cooked" meal? Or, what if your budget is tight and you can't afford to bring lasagna to a family of 8? We'll answer those questions in our next installment, so stay tuned! There's a solution for everyone.