Assigned seats on Italian trains?

If it's a Freccia or Intercity train yes. No assignment for regional trains.

You may select your own seat if you like. Make sure you have that option checked early on in the process or you will be assigned seats.

If you are buying "normal" tickets (not using a pass), the reservation comes automatically with the ticket if needed, it is not a separate purchase.
For a Freccia etc. the ticket will have train and seat details on it. For regional trains not, the ticket is valid for any train in the time window on the ticket, sit anywhere.

I hope you are buying at the Trenitalia (Italian National Railways) website: http://www.trenitalia.com/tcom-en

For a beginner's guide to Italian train travel see here: https://www.seat61.com/Italy-trains.htm

If buying tickets on the Trenitalia website, at the bottom of the page where you select the type of fare (base, economy, super economy, etc.), just above the Continue button, there's small box you must check in order to select your own seats. If you fail to check the box, the website will assign seats. If seats are assigned and you don't like them, just back up and restart the puchase process.

Thank you for your great tips! The seat61 website is a great resource! Since we know exactly which day and times we want to travel, I am inclined to purchase the tickets online from trenitalia.com as I am also able to select seats on that (never found that option on italiarail.com site.

So, my next question is how do you validate your ticket if you purchase it online?

You don't validate ANY tickets purchased online, they won't fit in the machine and are pre validated.