First time trying to buy train tickets - a little confused

You'll save on the fares on the fast trains if you buy ahead. That'd probably be #1 and #4. If there is a 5 ticket maximum, just order the tickets twice x 3.
If you happen to be close to the Florence train station, duck in there out of peak traffic time and buy your tickets to Monteroni and then down to Naples a day or two early. It'll simplify your departure to just figuring out which track your train will be on--and not having to fight with a vending machine for tickets.
If you want to do good reading on Italian train system, look at RonInRome.com. He even has detailed directions and pictures of the vending machines when buying tickets.

David, thanks for the plug and good advice!

I would only add that you're correct Nadia that you can only buy 5 tickets at once. If you have children (under 14), I would buy my tickets in bundles of 4 and 2... and I'd have a child in each purchase. That way you may qualify for the Family Rate on each purchase. When you buy the tickets this way.. in the SEAT SELECTION phase, choose the "sit near" function or select your own seats.. and odds are better you'll get seats next to each other.

DON'T LET TRENITALIA PICK YOUR SEATS! And if Trenitalia somehow messes you up (which they are known for doing), at least you have two folks together and four folks together -- and not one person hanging out in another coach by themselves! I buy tickets for folks this way all the time and have a very high rate of success.

For your Florence to Venice run (Really - the same day??? That's 4+ hours in the trains...) you could also look at Italo. They have limited trains running, but it gives you another option on that route. Because they can't come into Roma Termini, I'd not recommend them on the Naples to Rome route. Good Luck!

Purchasing in advance gives you a chance to save as the discounted tickets (economy or super economy) are limited in number and sell out quickly (sale starts 120 days before).
However you must realize that discounted tickets are non changeable and non refundable, therefore you must be reasonably certain of your travel plans because if you miss that train, you lose your money.
If you are not interested in discounts and want total flexibility, then you can buy tickets on the spot, seats don't run out as trains are frequent (2 per hour on average on your routes) and each train carries close to 500 passengers.
Florence to Venice costs €45 pp one way (full base price 2nd or standard class). You can save 25 to 60% by purchasing discounted tickets in advance. That is probably the segment with the highest saving potential.
Florence to Monteroni is regional trains and you can't get discounts nor purchase in advance. Buy on the spot.
Rome to Naples is also fast trains and you can save something, but not as much as Florence to Venice. Assuming you go from Monteroni to Roma via Chiusi, on regional trains, you can't save there, but you can save on the Rome to Naples portion.

Thank you for your advice - so helpful. My daughter is 10 and so she qualifies for a child fare.
I guess a day trip to Venice from Florence isn't very realistic? I don't want to wear ourselves out but was hoping to spend a day at least seeing some parts Venice.