I really love visiting the homes of my international friends. But I'm limited in the number of houses and apartments I can get into, with my wheelchair. I have a friend who is very allergic to peanuts, so going into a house where peanut oil is used liberally in cooking is not a good idea for her. There may be other obstacles or health concerns that limit the time you can spend in the home of your international friend. But just like speaking different languages, this does not need to be a barrier in your friendship. Inviting them into your space is just as important as being in theirs... and this is something I absolutely love to do.
When an international visits your home, you are the native, and they are the foreigner. So think and pray ahead of time about ways you can make them feel as comfortable and welcome as you possibly can. (This is when visiting their house first can really be a benefit!) Maybe have a comfortable seat available to offer them, and prepare something to drink. I like to keep my pantry stocked with tea and tea cookies! Here's a story about being prepared to be hospitable. Is there something kind and special you could do to honor them? It may be as simple as greeting them at the door in their traditional manner, or turning up your thermostat a couple degrees warmer than normal.
When an international visits your home, they become the learner, and you have the opportunity to be the teacher. This can be a very special privilege for both of you! I've had international friends over for specific things like a traditional American birthday party and Thanksgiving dinner. These are fun times to explain and demonstrate our cultural traditions. If there isn't a "special occasion," make it special by doing things you or your family love. Make a favorite meal like pizza, cheeseburgers, fried chicken, or macaroni and cheese, and introduce your guest to a favorite past time, like a board game, tag-football in the backyard, jigsaw puzzles, Charades, or a TV show. If you normally say a prayer before you eat, explain this and do it. While your routine might seem normal to you, you might be surprised at the response you get from your guest. It is special and unique because it is a glimpse into your life.
It's hard to know exactly how to strike a balance between making them comfortable and giving them a cultural experience. I don't think you need to go so far as eating your meal on the floor with your hands, or trying to prepare some complicated ethnic food. Remember, they have come to your space, to experience how you do things! Wouldn't you feel a bit disappointed if you visited an international home and they offered you a pepperoni pizza from Dominos?
I do recommend that if you are making a traditional American meal, that you include one dish that is a little more familiar to your guest - rice or greens or something. Steer clear (at least in the beginning) of making casseroles where all the ingredients are already mixed together; this makes trying new things intimidating! Don't serve pork or Jell-O until you know your friend well enough to know these are ok, based on their religious background. In fact, it might be a good idea to ask beforehand if there are any foods they are not supposed to eat. As your relationship develops, you could even get together to cook a mixed meal to share. One year for Thanksgiving, we had mashed potatoes, sweet tea, corn, wontons, and sambusa! Here's the story, if you're interested.
Because they are the learner, be prepared to answer a lot of "why" questions... try to anticipate some of these questions ahead of time, and think through good answers, instead of saying "I dunno... that's just what we do." Also take a look around your space through the eyes of your international friend. How does it look different? What do your decorations or cleaning habits say about what you value?
Time is often something we struggle with cross-culturally... If you say "Dinner is at 6," your friends may come early or late, and may hang around a while. In the beginning, meals with short prep work might be best, so as to avoid feeling frustrated over cold dishes or kitchen duty when you want to entertain. As your friendship develops and you gain more mutual trust, you can teach and explain the importance of time sensitivity in American culture. You might be learning to be flexible, but many other Americans will not be so understanding, so this is an important lesson for your friend to learn.
Finally, please remember that hospitality is much more than pulling off the perfect event. Do not get so wrapped up in the plans and preparation that you forget the goal: building bridges and breaking down barriers in cross-cultural relationships. Do this because you love the Lord, and because you want to show love to your international friend.