We're not commanded to give to The Church. Giving to the poor is giving to God. Giving is about demonstrating our thanks to God for providing for us in the past and our trust in God to provide for us in the future.
In what way are we commanded (or even urged) to give?
I don't see a command in the NT mandating details (quantity, timing, etc.) of giving. Some attributes of the method are given, but certainly not anything regarding quantity. The object of our giving isn't necessarily prescribed either. I certainly don't see a precedent for saying that we ought to give our money to The Church.
The closest example I see of giving as it's typically done in American churches is found in 1 Corinthians 16.
1 Corinthians 16:2a (NASB) On the first day of every week each one of you is to put aside and save, as he may prosper,
This instance is speaking about a specific need that the saints in Jerusalem had. I don't think this is the same as "giving to the Church." The local assembly of believers was giving the money to help starving people in Jerusalem. You could perhaps argue that Jerusalem is where the Church leadership was centered, but I don't read the text as emphasizing any special use the leaders had for the gift. It seems more likely that they would just immediately hand the money to one of the deacons (i.e. "servants"; Acts 6:2b (NASB), "It is not desirable for us to neglect the word of God in order to serve tables.") and not use it for any special "Church work" or administration.
Paul was the recipient of some gifts, and said that supporting apostles (maybe even evangelists?) was acceptable.
Philippians 4:16 (NASB) ...for even in Thessalonica you sent a gift more than once for my needs.
Again, I don't see this (and perhaps I'm predisposed to think so) as supporting the Church as a whole, but a specific individual or a specific need.
Neither is tithing a command under the new covenant. At least, it is not said to be so by Jesus or the NT authors.
How should we give?
Secretly—or Not Ostentatiously
Our motives should not be for selfish glory.
Matthew 6:1-4 (NASB) Beware of practicing your righteousness before men to be noticed by them; otherwise you have no reward with your Father who is in heaven. So when you give to the poor, do not sound a trumpet before you, as the hypocrites do in the synagogues and in the streets, so that they may be honored by men. Truly I say to you, they have their reward in full. But when you give to the poor, do not let your left hand know what your right hand is doing, 4 so that your giving will be in secret; and your Father who sees what is done in secret will reward you.
Also confer: Acts 5.
2 Corinthians 9:7 (NASB) Each one must do just as he has purposed in his heart, not grudgingly or under compulsion, for God loves a cheerful giver.
2 Corinthians 9:6 (NASB) Now this I say, he who sows sparingly will also reap sparingly, and he who sows bountifully will also reap bountifully.
There's no limit on how much (or little) we're supposed to give.
Luke 21:1-4 (NASB) And He looked up and saw the rich putting their gifts into the treasury. And He saw a poor widow putting in two small copper coins. And He said, "Truly I say to you, this poor widow put in more than all of them; for they all out of their surplus put into the offering ; but she out of her poverty put in all that she had to live on."
In fact, the greatest value in giving is in the quality of the sacrifice on our parts.
Philippians 4:17 (NASB) Not that I seek the gift itself, but I seek for the profit which increases to your account.
This harmonizes with what Jesus said about the widow who gave a small amount of money.
Acts 20:35 (NASB) "It is more blessed to give than to receive."
Like nearly every other requirement of Christianity, giving is about our hearts. Do we give freely out of generosity (without selfish motives)? Do we give out of love for God or someone else? These are the essentials of Christian giving.
God doesn't need our money. He doesn't seem to care about balancing his bank account. In terms of what he can do, there's always "more where that came from," no matter how much he has already given.
I don't see the Biblical authors prescribing the object of our charity, nor do I see that we have any sort of minimum standard of giving to maintain.
It's very simple (though difficult): Everything I have is God's to do with as he wishes. When I know of a need, I ought to do something about it.