Seeking help with the last minute Italy trip

1. What would be a frugal way to travel from FCO to Royal Marcela hotel in Rome? We are two adults and a 9 year old child.

Assuming you are staying in central Rome, the taxi fixed fare from Fiumicino Airport is €48 (no need to tip).

The direct train from FCO to Roma Termini station costs € 14 per person older than 12.

The FL1 train from FCO to Roma Tiburtina, Roma Trastevere and other secondary stations costs € 8 per person older than 12.

There are also many buses run by competing companies: https://www.adr.it/web/aeroporti-di-roma-en-/pax-fco-bus

including museums that one should book in advance.

All museum tickets should be bought in advance (on the official sites) as long as one doesn't like waiting in line to purchase tickets. On the other hand, ticket and security lines are quite short these days.

Advance Reservations are required to visit the Galleria Borghese, the Domus Aurea and some areas of the Coliseum.

will appreciate advice on restaurants in ... Vatican City

The only way to eat inside the Vatican City is to take the vows or to become an homeless. Tourists should avoid restaurants and cafes close to the Vatican like the plague.

All around Italy, You'd better avoid also restaurants that do not post the menu outside (with prices!) and those where they need a smiling guy on the door to get customers.

Seeing that your hotel is only a 5-10 minute walk from Termini, that would clearly be the cheapest option. However, if you don't pack like RS (and even if you do), I'd personally opt for a taxi from FCO as there are three of you the price per person difference is nominal.

Would we need to pay for Colosseum if go on the first Sunday?

According to the coopculture website (official ticket vendor for this attraction), yes. The site doesn't yet note any "free" days for this year. Also, if Sunday, July 4th was free, they wouldn't be selling timed-entry tickets for that day....which are, BTW, mandatory to gain entrance. See this page if looking for general-entrance tickets (click ahead to July and choose an entry time with tickets still available):

https://ecm.coopculture.it/index.php?option=com_snapp&view=event&id=3793660E-5E3F-9172-2F89-016CB3FAD609&catalogid=B79E95CA-090E-FDA8-2364-017448FF0FA0&lang=en

Your child, however, will be free: choose full admission tickets for each adult, and a "free" ticket for your 9 year-old.
More info here;
https://www.coopculture.it/en/colosseo-e-shop.cfm

If you are going on to Florence on Monday, note that Italian State Museums - such as the Uffizi and Accademia - are closed on Mondays. The Vatican Museums, BTW, are closed on Sundays, as are the restaurants within the museums; there are no other public restaurants within Vatican City. You can visit the basilica for free, though; only requires getting through the security-check queue.

Editing to add: have you purchased a good guidebook? If not, I highly recommend that you do so as it'll help you choose what to see/do according to your own interests. The only caution is that open hours, entry fees, etc. may have changed since they went to press - things are fluid what with re-openings after COVID closures - so you'll want to reference the official websites for attractions as well.

Trattoria Mario in Florance, is one of my favorite restaurants. It’s only open for lunch and is quite popular, so get there early. It was suggested to me by a local the many years ago I visited and it did not disappoint. I would also look into getting the Roma pass as it includes free entry into a museum of your choice and discounts on additional museums and attractions as well as free use of public toilets and free use of the public transport system. It also gives you fast track entrance into the attractions you choose. I believe Florance has a similar pass that might be worth looking into and pricing out the value for what you want to do. When in Florance, you can hop an hour train to Pisa and spend the morning seeing the tower and be back in Florance by lunch. No need to buy train tickets in advance and there is a train leaving every hour just about.

I believe Florance has a similar pass that might be worth looking into

Christina, that would be the Firenzecard, and it's currently not covering access to any museums.
http://firenzecard.it/en

A prior, timed-entry reservation is mandatory to enter the Colosseum even if using the Roma Pass. Reservations are also mandatory or highly recommended at some other museums in Rome covered by the pass as well. For details/instructions, see Section 7: Reservations under the FAQs on the Roma Pass website:

https://www.romapass.it/en/FAQ/#faqq38

As I'd mentioned above, lots of things have changed - either temporarily or not-so - with the COVID situation!

If you have a smart phone, I encourage you to download the free "Rick Steves Audio Europe" app. This app gives you access to MANY audio and visual walking tours in Rome, Venice and Florence. You can download these before you leave on your trip, and then play them (with earphones or earbuds) as you are visiting many of the sights in these cities. For example, two years ago my sister and I each downloaded "Rome - Heart of Rome" into our own phones and each of us listened to it as we walked through the central part of Rome together. In this particular tour, you can listen to Rick telling you about Campo dé Fiori, Piazza Navona, the Pantheon, the Trevi Fountain, and many other sights along a suggested path. He also has free downloadable tours of the Colesseum, etc. This is a very economical way to take tours on your own schedule, with Rick as your tour guide. With your child along, you could each wear one ear bud if your child doesn't have a phone. I've seen other people do this in order to share what's on one phone.

Also one good and economical place at which I've enjoyed eating a couple of times when in Florence is "Leonardo Self-Service." It's a cafeteria about 3 blocks from the Duomo and Baptistry; address is Via dei Pecori 11. Not well marked; look for a stairwell, as it's up on the second floor of a building. Here's one recent review: "Great place with delicious real Italian food for cheap prices. Get a tray and walk through picking out the plates of food that you want. Pay once you get to the end and walk on over to a table to enjoy it. Put the tray away when you're done. Super simple and easy. I had a big meal with 3 plates which cost me 10 euros. You could easily have a filling meal with 1 or 2 plates which will run you 4-8 euro. Highly recommend!"

These are probably the cheapest ways to eat in Italy: Buy food at supermarket, buy food at a street market, buy prepared foods at a deli (this will usually be sold by weight, so you must be careful not to over-order), or go to a pizzeria. I've had mediocre slabs of pizza from bakeries (stuff that has been out of the oven for a while), but I've never had a bad pizza from a pizzeria.

In restaurants you are not expected to order every course. I nearly always just choose two. Quality is generally very high in Italy, so I think it's worth having a few real restaurant meals. Gelato in an affordable treat. The cones/cups are not really large and thus are not terribly expensive. It will be a lot cheaper from a gelato stand than from a cafe where you sit down.

Be aware that you will pay for bottled water in restaurants; if you don't want the fizzy stuff, be careful to make that very clear, because the default assumption seems to be that you want bubbly water. If you want still water, check the label on the bottle before opening it. Budget tip: Try not to walk into a restaurant thirsty; that can really run up the cost of an inexpensive meal. If you're all drinking water, ordering a large bottle will be cheaper. You can probably get tap water (perfectly fine) without raised eyebrows if you're also ordering wine or soda. Cold bottles of water are widely available in little corner shops and in bars. Shops will usually be cheapest, but I remember paying only about 1 euro for a cold bottle of water in Puglia in 2015; it's probably a bit more expensive in Rome. Anything consumed standing up at a bar (water, coffee, soft drinks, beer) is likely to be much cheaper than if you consume it while sitting at a table.

For your 9 year old, there is a big public pool in Florence in the Cascine Park, just a little out of the centre.
Florence will be baking hot ,as it's in a valley so holds the heat in a sort of bowl.
It could be a great relief to have a swim!
I don't know if the pool is open yet, but check to see, and you may have to reserve a time.

And you must ABSOLUTELY have booked accommodations with AC.
You will roast otherwise.
Don't even consider not having it.

https://www.visitflorence.com/what-to-do-in-florence/pools-in-florence.html

This website has 2018 info, but it should be easy enough to find updates.
I don't think your little fellow will be too happy trailing around in the heat; but you may be from somewhere where you are used to it with high humidity?

Do you gave a good guidebook? These are the type of questions I look up in a book before I travel. It's good to read one thoroughly as you don't know what you don't know...

Sightseeing itineraries here:
https://www.ricksteves.com/europe/italy

The frugal way would be to take the Leonardo Express with a big BUT. It’s only going to be frugal if you’re able to walk from there to your hotel. If you have to add a taxi from the station to your hotel you’ll marginalize the savings and you’d be better served to take one from the airport. It’s not a long distance but it’s not a straight route. Are you going to be befuddled after a long day of travel? Will you child be up for a walk that may be hot? You’re probably tech savvy enough to follow the route, but sidewalks are narrow and uneven.