With 7 days I would pick one city.
With 10 days two cities ... maybe.
With no other guidance and it being a first trip for many I would first research London and Paris and Rome: These are the classics of American tourism in Europe and for a good reason.
Research? The DK Eyewitness Travel book for each. The RS book is great for planning once you decide but too clinical at this stage. Then visit YouTube and start watching tourist videos like this Paris video https://youtu.be/EEeu7-xJX_c?si=5Mzj8ISSrLIrY7xe Nothing works better for me. I only do the short ones and that enough. Collect up three favorites from each city and share them and ask for a vote.
The budget. There is this website. I don’t know how up to date in accurate it is, but maybe to develop an initial understanding of what this might cost you … maybe it will be useful. https://www.budgetyourtrip.com/italy/rome I often help travelers coming here budget their trip and I presume that if you were to post in the Italy, France and England forums someone there will help as well. That website says for a mid-range week in Rome costs $3,108 for two people. This includes accommodation, food, local transportation, and sightseeing but not airfare; and that sounds about right for the summer. For the airfare have $1200 in the back of your mind until you narrow down the destination and can check prices. The same site shows about $1500 in Budapest for a week and about $4000 for London; just to show you that there is no "how much does it cost to go to Europe" answer.
If you can’t afford London or Paris or Rome, dont try. There are magnificent but less expensive destinations where you can have wonderful holiday ... but thats only good if the location interests you. If you get down into checking out those three cities and the budget gets busted, then look a bit further east. A 7-day trip, maybe Prague https://youtu.be/ffxGB2j9ADo?si=MVfXBRhkFlbWhz5l or Budapest https://youtu.be/1nd5AtZIrTk?feature=shared AND https://youtu.be/0nd9DuDGCz0?si=aa0c19KaJHEtabkXor Krakow https://youtu.be/FAdEdA1-l6M?si=tvNQXw4nUIc9ptT9 . A 10-day trip, two of those. Or while Vienna isn’t cheap, if you combine it with Prague or Budapest and it balances out a bit.
The last thing I will say is dont compromise. This is a big deal, and you want it to have the full impact. If, because of cost, to do 10 days you have to stay in a 2-star hotel a 45-minute tram ride from the center, dont. Its not always about quantity, sometime you gotta put quality first. Instead do 7 days and stay in the heart of the experience. You might be told that for a short trip, first look for direct flights. No, not really. Sure that’s best if its to a place that you will remember for life and its affordable. But I guarantee that if you have a great time on holiday you won’t remember the change of planes and the extra 3 hours to get there.