Speaking from personal experience, I recommend that you research what ministries or non-profits are already present in your city. They may function with a small staff and a small budget, but they are often the ones who have the best personal connections with the international population in your community. When I lived in Fort Wayne, I served with International House, which has been serving in the same area for about 15 years, and native Hoosiers are still shocked to discover it, right in their backyard! You may be surprised to learn that there are people in your area who are already providing services for international people. A quick Google search of your city, with key words like "nonprofits and charities" or "refugees and immigrants" could be a good place to start. And of course, don't underestimate the leaders in your church as a possible connecting resource.
There are also bigger organizations that have multiple offices around the country and the world. World Relief and Catholic Charities are just two of the well-known, well-established non-profits that can be found in many cities. Check out their websites below to see if they are near you:
2. Resource centers
Many social/community resource centers help international people, such as free medical clinics and language learning centers. The Red Cross, Salvation Army, and the YMCA are also places where families will go for support. You may want to volunteer or find part time jobs in one of these places.
3. Places of work
Get out of your comfort zone and eat or shop somewhere you don't normally go! Authentic Mexican restaurants, Indian buffets, Chinese take-outs, or Japanese sushi bars are excellent places to try some delicious foods and meet some interesting people. Also keep your eye out for local ethnic markets or boutique shops to browse and shop in. If it's the real thing, then chances are it's a small family-run business, and the people who work there are from that country. When you find a place you particularly enjoy, make an intentional effort to frequently visit. The owners or servers will start to recognize you and will be more friendly and open to conversation.
In my hometown of Winston-Salem, there are churches on every corner... sometimes two or three on a corner! I notice that some of the signs/marquees are in Spanish or Korean. I also know that there is usually a variety of nationalities represented in Greek Orthodox churches. Pay attention to the signs of churches in your community. Take note of the ones that are written in other languages or have international flags displayed, and maybe skip your church service some week to visit one of these churches. Be prepared to not understand a word that is spoken or sung, and be open to experiencing styles of worship that might look or sound different from what you are used to. This is ok - it's a cross-cultural experience! People will be very curious about why you are there, but will also be extremely welcoming to you.
Do some research, explore a new corner of your town, be an observer and learner... and pray for God to open your eyes to see the nations among us!