Are the Word and Spirit also called God because the Father isn't made up of parts in accordance with the the doctrine of the Trinity? Do Christians worship the Word and Spirit as God too because the Word and Spirit are inseparable attributes of the Father and because the Father isn't made up of parts? Actually I am just trying to understand the logic followed by the early church fathers in declaring God as a Trinitarian being. The Jews worshiped only the Father as God. The Word and Spirit were declared God later by the followers of Christ.
Yes, Christians worship the Son and the Spirit as well as the Father. 

The Son (the Word) and the Spirit are as much God as the Father is. At any given time a given Christian may be focussing on worshipping only one of the Persons of the Trinity, such as singing a song of worship specifically to Jesus, but none of the Persons are excluded from worship.

Whether the Jews worshipped only the Father is a more difficult question, and it's more about semantics than it is about theology. A Christian would say that by worshipping God the Jews were in fact worshipping the whole of God, in all three Persons. Jews would not see it like that, since they would not believe in the existence of God the Son or God the Spirit. But they would say that they are worshipping 'the whole of God', which to a Christian would include the Son and the Spirit.