www.renfe.com is the site of the main train operator in Spain.

If that is the order you're going in, then that is the only operator you will use. There is competition on Madrid - Barcelona and Madrid - Valencia but looks like you won't be on those routes.

For Málaga to Valencia, I would recommend flying. Ryanair flies it year round and Iberia flies in summer season. (might be a good way to spend some avios if you have any)

From Valencia to Barcelona it's the Euromed which isn't true high speed but it's nice and fast enough.