Some apartments are listed on booking.com, but there are other websites that probably offer more. Others here can give you specific suggestions.
You can get from La Spezia to Rome in 3-1/2 to 4-1/2 hours, depending on which departure time you choose, so you don't have to even think about an overnight train. (Some may be reasonably comfortable, but I had a nightmarish, no-sleep trip from Rome to Sicily in 2015.) I find the Deutsche Bahn website easiest to use when checking train schedules in Italy, but to get fares and purchase tickets you'll need to use TrenItalia, which will insist that you use the Italian spelling for each origin and destination (i.e., "Roma", "Venezia").
Edited to add: When you're certain you have the date and rough time of your La Spezia-Rome trip pinned down, I'd look at the departures around your desired time and compare length of trip, number of changes (there are a few that require no change at all), length of transfer time, and cost. If I were moving a family around, I'd be willing to pay a bit more to reduce the number of changes I had to make, and I would try to avoid really fast changes in large stations, though it will help if you all are traveling light and everyone can manage his/her own suitcase.
2nd Edit:
It appears that you will be moving from La Spezia to Rome on Friday, June 2. TrenItalia doesn't have those trains loaded yet, but the schedule probably won't be terribly different from what I see on Friday, May 5 (though most of those trains aren't being sold yet). You can look at the trains available now and make a provisional selection, then you need to keep a sharp eye on the TrenItalia website so you can buy your tickets as soon as they go on sale if you are looking for the cheapest non-refundable/non-changeable tickets. There's nothing wrong with waiting to buy if you're unsure about your travel schedule and prepared to pay more (could be 3 times as much or perhaps more??).
These are some of the good-looking options on the schedule for May 5:
Dep 8:16 AM via Frecciabianca, arr Roma Termini 12:03 PM. Great price of 19.90 euros on this run that requires no changes! By staying on your toes, you might snag June 2 tickets for that price. (Kids' tickets may cost less; I haven't checked on that.)
Dep 10:06 AM via InterCity, arr Roma Termini 2:33 PM. Another option with no changes. The IC train is a bit slower than the Frecciabianca and the May 5 train is currently selling for 12.90 euros.
Dep 1:16 PM via Frecciabianca, arr Roma Termini 5:03 PM. No changes. Price for May 5 is 29.90 euros; guess people like travel in the early afternoon on Fridays!
Dep 1:52 PM via InterCity, arr Roma Termini 6:03 PM. No changes. 19.90 euros for May 5.
Dep 3:40 PM via Regionale, change in Florence (31-minute layover, not rushed) to Frecciarossa, arr Roma Termini 8:10 PM. This connection isn't priced yet, but it's an example of what might not be a smart ticket to buy. The Frecciarossas are expensive trains, but you'd be losing the time advantage you're paying for by spending 31 minutes waiting for that train in Florence. You'd do about as well, if not better, with other combinations that likely would be less expensive, though you won't know for sure till prices are posted for June 2.
For example, dep. 4:16 PM via Frecciabianca, arr Roma Termini 8:03 PM. No changes. Likely to be cheaper than the 3:40 PM departure, leaves La Spezia later and arrives in Rome earlier. This one's available for May 5 at 29.90 euros.
There's not anything wrong with the other options; just be sure any connection times are not scary-tight. It's not a bad thing to have to change trains. It's not hard, and it's useful for the various members of your family to see how easy it is. You should feel free to select a departure time that fits your schedule for the day.