I've been to all three, but out of them spent the most time in Paris, so I'll focus on that.
For Paris, the metro makes getting around super easy and convenient. There's also a weekly pass available called
Navigo Decouverte, which for some reason isn't well advertised. But it does require a slightly smaller than passport sized photo to get. This also gets you to Versailles and to the airport if that's relevant, so it's a pretty good deal (pays for itself just with a round trip to the airport). Sadly, it's only valid on a single Monday-Sunday period, so if you arrive on Saturday and depart on Tuesday, you'd need two passes which probably won't be worth it (and I don't think the only one you could even buy on Saturday would be for next week).
Read up on common scams. This is good advice anywhere, but Paris in particular was filled with people trying to get you to sign random petition sheets and the like. I was even hassled at the airport of all things. Just ignore people like that and they'll go and hassle someone else.
Also important: if you're an EU citizen and under 26, basically all cultural attractions are free to enter. Including the Louver, Versailles, d'Orsay, and so on.
I don't know if this changed recently, but when I was there everything had security checkpoints. I'd travel light so that you don't spend too much time getting through them.
Sights wise, you're not going to be able to see absolutely everything in 4 days. So I'd suggest looking things up on tripadvisor/wikivoyage, and prioritizing based on your interests.
If you're into art, Louvre is obviously great, and so is d'Orsay. I'd recommend getting the audio guide for both, since it adds a ton of info to the visit. And the one in the Louvre also comes with a map, which you'll likely need because the place is huge. Just note that the Louvre one requires a piece of ID as a deposit, so make sure to carry one with you (or if you're using the free admittance mentioned above, carry 2, as you'll need one to actually get in the museum). The Louvre also has a side entrance that is supposedly less crowded, but I went there in November and just walked in the front door. Pompidou is also pretty cool, even if you aren't super into modern art.
For churches, Saint Chapelle is incredible, but can be a bit tricky to get into because there are two lines for the building it's in. One for the actual court house, and one just for the Chapel. Don't be like me and spend time waiting in the wrong line. It's also closed between 1pm and 2.30pm every day.
I found the Eiffel Tower to be pretty disappointing honestly, but it's still something you want to go to just to say you've been there. Just don't get your hopes up too much I guess. The touristy restaurant in the middle of it is also pretty crap, although not all that expensive for lunch given its location. Champs Elysées was also pretty disappointing to walk down, and I generally love walking down main streets in most European cities.
In terms of food in general, the best food I ate in Paris was in super small local restaurants. But it was generally all good but expensive. I also don't have personal experience with them, but there's apparently a bunch of pretty cheap Michelin star restaurants in the city if that's your thing.
For Rome, I was only there a couple of days, but the Vatican Museums and St. Peter's Basilica are absolutely a must. For London, a ton of museums have free entry, and a lot of them are great.
In all three cities, "free" walking tours exist (not really free, since you absolutely should be tipping the guide). I like taking the first day in a new city to go on one, since they generally give you a pretty good overview of the city and where you might want to go next. But on the other hand, you're pretty limited time wise in Paris and London, so they might be a waste.
