A few weeks ago we featured the book review of "The Gospel Comes With A House Key".  It is written by the super-insightful Rosaria Butterfield and is available on our Book Wall.  She makes the case that hospitality has been a beautiful reflection of the gospel from the beginning.  She also invites us to consider how we may be missing out on the blessing of community that result from hospitality being a more central part of our lives.  It helps us more deeply process what Paul meant in Romans 15:7 when he says "welcome one another as Christ has welcomed you".
Buy a copy and read it.  You won't be sorry!  
In the meantime, to add to the conversation, here are some suggestions to get started from Matt Chandler (Lead Pastor of Village Church in Dallas, Texas).

Four Ways to Show Hospitality

The God of the universe is serious about hospitality. It can create an entry point for living out the Great Commission and evangelizing our neighbors, especially in the age of unbelief.

How do we show hospitality today? It’s not complicated—though that doesn’t mean it’s easy. Here are four ways.

1. Welcome Everyone You Meet
I think the best thing to do is literally greet everyone you see. That’s an easy thing to do if you’re wired like me—I’m a grade-A extrovert. That’s hard if you’re an introvert, and right now you’re thinking, Can we just go to number two, please? But often the best things to do are the hardest things to do.

Pray for grace, ask for strength and, well, greet people.

2. Engage People
Remember that everyone you meet is eternal. You’ve never met a mere mortal, and you have never met someone who doesn’t bear God’s image. So care about and take an interest in those you run across. I don’t think this is overly difficult.  We simply need to ask open-ended questions and let our inner curiosity out.

You may think this is all obvious—but so often we hold back from doing it. You need to get to know people, take an interest in them, and listen to them rather than just trying to think about how you can say something memorable or hilarious.

3. Make Dinner a Priority
The Bible, over and over again, talks about the holiness of eating together. Long dinners with good food, good drink, good company, and good conversations that center around our beliefs, hopes, fears—that’s a good dinner. The Bible says that’s holy.

Oh, and I don’t mean dinner with friends. Yes, eat with your church small group, invite over your good friends, but remember that hospitality is to give loving welcome to those outside your normal circle of friends. It’s opening your life and your house to those who believe differently than you do.

4. Love the Outsider
In every work environment, every neighborhood, there are people who, for whatever reason, are kind of outliers. These men and women are all around you—perhaps more so than ever in our globalized world.

Because of the way sin affects us, we tend to run away from differences and from being around people who think differently and look differently than we do.

But I want to lay this before you: Jesus Christ would have moved toward those people. God extends radical hospitality to me and you. We love the outsider because we were the outsider.
(This is an excerpt from an article found at thegospelcoalition.org/article/hospitality-courageous)