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.
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