I have a smartphone question for you guys?

If I send a text from my LG Android phone 2 my wife's iPhone XR will she receive the message right away and will it pop up in front of her screen like my Android phone or just go straight to her message inbox and then she have to look to see if she had any messages

Depends on how her notifications are set up. If they are left as normal then yes it would come on the display (lock screen) for her to do whatever.