I realize that the Catholic chuch offers a Saturday night service that is in essence the same as the Sunday morning service. Some people go on Saturday because they are working or otherwise can not go on Sunday but still want to honor God and the sabbath.
I would go on Saturday if I could not go on Sunday.
Now, as to what I would expect, it would depend entirely on the congregation, denomination and motivation for going. If my motivation was to attend worship because I could not go on Sunday, I would want it to look like that service. If I was going because I had a problem with Sunday, was formerly unchurched, etc., I would expect those things that I took issue with to be missing or something new added.
I can tell you what I would like to see different about church and I'll preface by telling you that my context is that I love church, I believe that God is present among His gathered people and that there is no where I would rather be but you asked so here is what I would change:
1. I would want every service to be praise filled and relevant. I want to always feel like people leading worship, including the readers, believe what they are saying, that they believe God is present. Now before I need one really big pair of tweezers to get the plank out of my eye, I'll tell you that is much easier said than done...but there you go.
2. I would want vision, clear ideas of what was expected of me as a Christian. I don't mean perfection. I mean, "What does picking up my cross and following look like to me, personally."
3. I would want an opportunity to offer up my own prayers if I wanted to. If we are there to pray, why can't we all pray. If the Spirit is in all of us, why does only one pray?
4. I would want a clear bulletin and no stumbling blocks that prevent me from participating in the service. I don't want to feel like it is a members only club but a welcoming community of Christ.
Now if I were to go to a church for a first time, I can tell you what my number 1 thing that is an indicator if I will ever go back...I want to see the people look like Christ.
If they welcome me, make me feel at home, offer for me to sit with them, find me a bulletin, etc. I know that this is a place that raises up disciples not members. I know that they "get it" I want to know that there is a sense of community. I want to see that the people in this place help people in some way, preferrably many ways, that may not be members of the church.
I want a place where my gifts are recognized and can be utilized. If I can't contribute, I don't want to watch you or anyone else being Christ to thier neighbor, I want to go do it.
I want to encounter Christ.
Now if you can do all of that, not only will I move there and attend, I want you to train me on how to do it.
I don't think that you can be perfect or offer the perfect service at the perfect time. I do think that you can proclaim Christ crucified, be open and vunerable, pray that God will work through you in whatever way brings Him the most glory and get out of the way and let Him do it.
God's blessings to you in your endeavor.