I've now looked into this a bit myself. I found a program which will transfer and merge messages between an iphone and an android phone:
http://www.backuptrans.com/android-iphone-sms-transfer-plus-for-mac.html
I contacted the developer about a Windows Phone support. I was told they will be bringing in Windows Phone support but don't have an ETA yet.
If anyone knows of any other similar products, please do share.
Like what for example? (I don't use any such apps)
Do the people I am communicating with have to use the same app or does it take over default messaging functionality on my phone?
Does the app import old messages from the stock messaging app? (at the moment I am on iPhone)
Is this referring to transferring messages between different Android phones (i.e. not between operating systems)
A backup and restore process will be required to sync standard SMS messages from an Android phone to an iPhone and back, since due to the nature of SMS they're delivered directly to the device (not a cloud service).
There are several solutions that will sync messages across Android devices since background apps can be given permission to read/send SMS, but you can't do this on an iPhone.
An alternative is to use something like Facebook Messenger, WhatsApp, Snapchat or one of the other internet/cloud-based messaging apps.
If you're on Verizon, there is a carrier solution called Verizon Messages. It's an app that will sync SMS across devices though Verizon's systems provided you use their app, though at present it's available only for Android and iPad; there is no iPhone version since there's no way to sync with the built-in SMS/iMessage app on iOS (works on iPad since there is no SMS app it has to interface with). The app is not the prettiest, however, even on Android.
Unfortunately this is one of the big disadvantages to switching back and forth between iOS and Android - I have an iPhone and an Android phone, and I'd sometimes like to just pop my SIM in one or the other and go, but SMS syncing is the one key part that doesn't quite work without significant effort (unless someone knows a solution I do not). Everything else is simple (email, calendar, my Adobe Reader PDF files, Evernote...).
So unless you can get everyone who matters to use something like FB Messenger or something similar, text messages between iOS and Android won't be well synced. Android to Android on the other hand is simple via one of several third-party apps.