Hi m5fan,
I use an iPhone 5 and a Nexus 5 and both will have delays in the sending of text messages at times sending to both other AT&T devices and other carriers, maybe 20 to 30 seconds at times but not 2 to 3 minutes as you are describing.
Have you installed any other "texting" apps (Such as Whatsapp, Text+, TextFree, etc)? Have you activated the combined SMS/Hangout Inbox? These could all cause problems with native texting on the device and could be part of your issue.
Have you ran all the software updates on your Note 3 as well?