Can I forward messages received just like call forwarding?
Yes and no. With an iPhone you technically can forward your text messages but only to other Apple devices such as an iPad, iMac, MacBook, etc.
I have my text forwarded to other Apple products and this can be setup via: Settings > Messages > Text Message Forwarding.
In general, you can't forward text messages as you do calls. This is a great advantage of the iPhone that I use daily. The other option is to use Verizon Messages: Message+ on the App Store. This allows you to respond to texts via a PC etc. if need be.
Let us know if this helps!