I am looking for an app that can take all SMS messages received by my phone and forward them to my email address. I need it for Android 4.0+.
- Supports Android 4.0+: Yes, 2.3 and up.
- Forwards incoming SMS to e-mail: Yes, e-mail and other phonme numbers are supported
- Forwards all incoming SMS: Your choice. It even allows to forward messages from selected senders.
Screenshots (source: Google Play; click images for larger variants)
I'm not using this app myself (I rarely deal with SMS), but it has a pretty good rating at Google Play (4.2 stars at over 2,500 votes), so I feel confident this is a good recommendation. And other than SMS Gateway (which was recommended in this answer and had its last update in 6/2012), SMS Forwarder seems to be still maintained actively (last update 5/2015, as of this writing). For alternatives, just follow the first link to my list :)
SMS gateway does what you need and more.
It is essentially a gateway between SMS and other forms of electronic communication.
What you are trying to do can be done using the forward to smtp option. You would have to configure it with gmail or some other smtp server in the smtp settings of the app.