Acts 2:42- "And they were continually devoting themselves... to the fellowship..."
Hebrews 13:2- "Do not neglect to show hospitality to strangers, for by this some have entertained angels without knowing it."
Some may use the words fellowship and hospitality interchangeably but these are really different things. Fellowship is what Christian brothers and sisters enjoy with each other. It means sharing or having things in common. You may be of different ethnic make-up or different economic level or different personalities or different interests and hobbies but what you have in common is more significant and more binding. You are common recipients of the grace of God which has transformed you to new men and women. The church may be multi-ethnic or diverse in its personalities but it is not multi-cultural. The church represents one new gospel culture. The more diverse a congregation the more magnifying of how the gospel brings such diversity together in one new culture. Fellowship is more than socializing or having pot-luck suppers. Fellowship is demonstrated by the continual effort of every church member to invest relationally in other church members- to encourage others, carry each others' burdens, labor together in ministry, and celebrate together in worship.
Hospitality is a different animal though. It refers to your demonstrating kindness and assisting with needs of strangers. For Christians this means demonstrating godly character and kindness to those outside the church-- visitors to your local church, yes, but especially those who are strangers to Christ. The gospel is a message lived out. Outreach and evangelism are not programs on a Tuesday night that happen with only a small group of extroverts in the church. These are the mandates of every Christian and the gospel message is adorned with Christian hospitality. People who visit the local church ought to be extended kindness-- not just a bulletin. Go sit with them (don't ask them to get up and sit with you-- leave your spot and go to them). Take them to lunch. Make an appointment to have them to your home. Shock them with how generous you are with your time.
Hospitality cannot just wait for people to come to us though. Work on getting to know your neighbors. Bake a big batch of cookies (slice and bake are ok), bag them up, and deliver them with a kind note. Who turns down cookies? Follow that up with an invitation to your home or to dinner out. If you know of a need, offer to help. Find out birthdays and send cards. These things are not sharing the gospel in and of themselves but they may pave the way for an opportunity to share and when you share the gospel, the message will have been commended by your unique and Christ-like character. Don't wait for a flashy "new way we're going to reach people" program to get started at church (those usually have high costs and low returns anyway). Do outreach like the Bible shows us to do it and trust the Lord with the results!