Congratulations. Regarding your questions, in my answers, I will assume you don't have qualifying citizenship (EU, Schengen, EFTA), which would give you access to member countries like Italy as a non-visitor (and you'd be required to enter on such passport if you had one).
In 2025, US citizens don't need a visa for tourism in Europe (see #5): US citizens generally don't need a visa for EU tourism or business for 90 days within the previous 180 days. Schengen requires a US passport have 6 months of validity.
Proof of financial condition in EU: Generally, US citizens are not asked in Europe. If one is very young and gives off a suspicion of overstaying, I suppose it's theoretically possible that the officer may ask for ties to the US. I've never been asked for financial proof even when I was very young.
Letter from sister: I've never been asked for such letters in the EU even when visiting residents. Sometimes, the officer does not even make eye contact and stamps my passport without even a hello back.
Insurance is required for Schengen: They don't check but insurance is required for visitors. Be on the safe side and always travel with travel insurance even for EU countries that in practice don't always bill visitors. One time, my friend needed antibiotics in Rome and the doctor asked for him to go to an ATM and take out some cash. Each EU nation has a different healthcare system so there is not uniformity in rules. To be safe, US citizens should always expect billing for any care received even if a bill is not provided when discharged. Most travel insurance policies only covers emergency and catastrophic healthcare, medical transport (very expensive), and repatriation of remains (I hope not!) Since I travel a lot, I buy an annual policy of GeoBlue Trekker Choice international travel insurance, which can also be used for non-emergency care (e.g. an ear infection).
Bad news: in 2026, US citizens will need an ETIAS authorization to visit most of Europe. The European Union will adopt an ESTA-like system for visitors called ETIAS in 2026. Once implemented, Americans must apply for a ETIAS authorization and pay 20 Euros via a website. An ETIAS authorization will be valid for 3 years or the expiration of the passport, whichever comes first.
Good news: in October 2025, EU will allow US citizens to use e-gates at passport control: According to this EU press release, EU will allow visitors to use e-gates but American visitors must see an officer on their first entry. Some passengers may be pulled aside for manual inspection.
Where do US citizens need a visa? Wikipedia has a great resource to show you what the visa requirements are for US citizens throughout the world. It's generally a good idea to verify the requirements on each country's official immigration website. Enjoy seeing the world on your new US passport.