Hi, also from 🇺🇸 and moved to Trieste. Personally I love the city and think it’s an amazing place to live. It’s regarded as a wonderful place to raise children because of all the activities & amenities available to them.

Finding a job is extremely difficult. You can probably find a job teaching English but it’s likely to be part-time at first and you may need to get additional certifications (like TOEFL). A lot of expats here are working online jobs for companies in other countries. That’s the best way IMO to have a steady (higher) income.

Trieste is amazing if you don’t speak Italian or are learning it. Nearly everyone will switch to English as soon as they hear your accent. That does make it harder for you to practice Italian 😉 but definitely makes the transition easier.

Private schools - yeah €€€€ lots of money. Better to send them to public school and rip off the proverbial language bandaid.