If I graft two trees together while young, will they grow as one plant?

If two trees grow close enough together so that their trunks touch each other anywhere along the length of the tree, then they will eventually fuse. This generally only happens at the trunk because, unlike small branches, the trunk really can't be pushed out of the way as easily. It doesn't necessarily need to be two trees of the same species either.

There used to be a fused sycamore-maple on my school campus (it was damaged in Sandy and was cut down). They weren't completely fused together, but you could see a joint at the base and about 20 feet below the canopy where the trunks essentially became the same. There was no distinction between the two separate trunks.

But to finally answer your question; when the trees fuse they pretty much become conjoined twins. I'm not sure if they transfer genetics to each other, but they do share resources.