What does it mean to "worship God" to Christians in general? When Christians worship or pray to God, generally what is in their mind? Do they pray to God who is one person among Trinity or do they pray to all three of them as one?

I am asking about what is in their mind in general, not what should be in their mind theoretically.
In the Nicene Creed, one of the oldest formulations of the Faith, and one which is repeated every Sunday by a large portion of Christianity, it says of the Holy Spirit:


  I believe in the Holy Spirit, ... who with the Father and the Son together is worshiped and glorified, who spake by the prophets


In theory then, all three members of the Trinity are worshipped and glorified equally.

That said, your question specifically asks for practice, not theological formulation.  And, since the Trinity is a hard concept to grok, one should expect significant deviation in practice.  That being said, both in my Baptistic and Episcopalian circles, it is not all uncommon to begin a prayer with "Father God Almighty" (I do) and to end with "All these things we pray in Jesus' name."  That one should so easily morph from one to the other speaks to the fuzziness of the distinction in our minds.

When discussing the role/s of the Godhead, study leads us to separate the individual persons of the godhead.  Reduction is a useful thing in study - that's why we dissect things.  That said, when you cut something up, sometimes you kill it :)  I say that to say that my formulation of prayer (again, not uncommon) shows how we can dissect the roles, but still see the sum "Three-In-One" as one being.  

Mormons and other non-Trinitarians (a very small percentage of Christians) very clearly believe God the Father and God the Son to be two separate people - but those of us who are Trinitarian tend to be a little wibbly-wobbly on the whole thing.  In practice, we have a lot of bi-natarian theology (God the Father zaps you, and Jesus' blood saves you. That Holy Spirit Guy? Only Pentecostals and charismatics even know who he is!) - but even there, most Christians realize that Jesus is correct when he says "I and the Father are One."  

I can only speak for me when I pray - I see the two fuzzily morphing in and out of oneness together - but I suspect that fuzziness is pretty common.