Is this sentence grammatically correct - There is a spoon and a fork on the plate?

Not technically: the compound subject uses the plural verb are.

"There are a spoon and a fork on the plate."

But colloquially this construction, while erroneous, is very common. It could also be grammatically correct if the spoon is not on the plate (set off by a comma).

"There is a spoon, and a fork on the plate." (and there is a fork on the plate)