First, my short answer: Yes, I do tithe... to my home church.
To add to someone else's earlier comment: most Biblical references to tithing are found in the Old Testament books Leviticus and Malachi. If I had a little more time, I could go through and jot down the various Scripture verses.
In most 'normal' (as opposed to the 'flashy' churches with the televangelists) evangelical churches, it is usually taught that one tithes to their home congregation first and foremost
before any extra offerings or donations both internally and externally. An internal offering is an additional donation given to one's home church beyond
their tithe; examples of this could range from a church building fund to supporting a missionary (or missionaries) with ties/connections to the local church. An external offering is a donation to a ministry or organization that is independent of the local church; examples of this are Campus Crusade for Christ, Prison Fellowship, Focus on the Family, the Billy Graham Evangelistic Association, etc.
Tithing to an external organization en lieu of one's home place of worship is usually discouraged. As a matter of fact, most respectable outside ministries make it a point to say that people should not donate to their organization if it means that they will not be able to tithe to their home congregation.
There is now an agency (the Evangelical Council of Financial Accountability or E.C.F.A.) that many ministry organizations can apply to in order to receive a level of certification to ensure the public that donations will be used in an honest manner. In order to receive such certification, there are certain minimum standards that the ministry organizations had to achieve and maintain. More info. can be found on the below web-link.
Worth noting: one known 'flashy' televangelist in recent years applied for such certification but was rejected because it failed to meet the ECFA's standards. So the system does somewhat work.
"TransEastern! You'll feel like you've never left the ground because we treat you like dirt!" SNL Parady ad circa 1981