If the kid is a very strong student, with a chance at the top schools, here is what I would do.
Research the schools in each city carefully, in depth, beforehand, including virtual tours, to help you to eliminate schools which would not be an option for kid. The ones that look possible, plan to take a student-led tour and make a visit to the admissions office (to demonstrate interest), but also, if the kid has any type of identity that would help them to connect at the school, plan for that during the visit. For example, if kid is LGBTQ, a particular minority religion/ethnic group, an aficionado of a particular campus activity, anything that could make the visit more than just a tour and walk around, plan ahead for a visit to that office/club, or ask for a match up with a student outreach person. If they know someone from their high school who’s already there, reach out to them to meet up at the school that day, to get an insider tour, maybe go to a class or two with them.
I’d start in DC, in the hopes of catching the last of the Cherry Blossoms in the Basin area (which, depending upon how cold the spring is, and how early in April you go, you’d still have a chance of seeing). Of course Georgetown, and if kid is interested in majoring in Poli Sci or Int’l Relations or anything national government/politics related; also GW and even American U as a backup, because its star is climbing, and it’s worth it, to be in DC to have easy access to all the political and international relations resources/opportunities. Aside from touring those two campuses, walk the Mall, see all the outdoor monuments, go into any of the museums there. Take a look at the list and plan ahead according to your interests. Our Museums, Galleries, and Zoo | Smithsonian Institution If you have a car, the Air and Space’s enormous annex out in Chantilly, Va is amazing, well worth the drive. But for a first visit to DC, there is just so much to do and see on the Mall. If they’re letting people into any of the government buildings, see those, too, but I don’t know if they’re allowing people in due to Covid. You can contact your House Rep ahead of time for tickets - but I bet all that is suspended now. College-wise, DC is only worth a day for you, but tourist wise, it’s worth as many days as you like. DC has a ton of great ethnic food because so many nationalities are there. My favorite food to eat in DC is Ethiopian food, since the city has a large Ethiopian population, the largest outside of the capitol of Ethiopia, so there are a lot of good Ethiopian restaurants there. Google maps restaurant finder is very useful. You’d have to get away from the Mall to find something good and reasonable to eat, but it’s worth the walk, metro, or cab/uber. With two of you, uber probably worthwhile, especially if you don’t want to spend your time figuring out the metro, which isn’t hard to deal with, but if you’re not used to urban transit, the metro might be intimidating for you.
I’m not sure that Baltimore would be a match for your kid, in terms of a city, but if so, then of course Hopkins.
If you’re considering looking at Penn, take Amtrak to get to Philly - quicker than the trouble of using the airports. Penn is a fantastic school with an urban feel, but a true campus (unlike some urban schools, like NYU, BU, and Northeastern, that are mostly a series of buildings strung out along a major city street), a short walk from center city, Philadelphia, and a possible walk to other areas of the city, which are of course also reachable by public transit. Penn definitely checks the box of top large urban school in a big city. Other than Penn, the major thing to see in Philly is Independence Hall and the Liberty Bell, iconic sites in the founding of our nation. Cool places to eat not too far away would be Chinatown or Reading Terminal Market. A less competitive but still very good school would be Drexel, right next to Penn, which was originally an engineering school, but has expanded greatly into other fields, too. One day for Penn, maybe Drexel too. I’d say that you can do both schools, and Independence Hall/Liberty Bell in two days. I would not recommend Temple, because it’s in a truly dangerous area of the city, not as selective, not necessarily a match for a very highly qualified student. Villanova is in a wealthy quiet suburb, not what your kid seems to want.
Reagan airport in DC is easy to get to, but the airports that serve NYC are far from the city. For this reason, you might consider flying straight from DC to Boston (there is a free quick bus from the airport to South Station, and from there it’s easy to get to Boston schools by public transit, maybe cab/uber. Beware - Boston traffic can be horrible, since, like many very old East Coast cities, the streets weren’t planned - some are literally paved over stock paths! In Boston area - Harvard, if kid has a shot or wants to see it. Northeastern, which has become a quite selective school, and BU. All big urban schools, all worth investigating. Ridiculous to try to see them all three in one day. Do Northeastern and BU one day, Harvard the other day. Aside from the schools, the thing to do/see in Boston is The Freedom Trail (American historical sites), as much of it as you have time for. https://www.thefreedomtrail.org/ , and the adjacent Public Gardens.
Princeton is not in a city at all, and Yale, while a wonderful school, is in a smaller city (New Haven) which cannot hold a candle to DC, Philly, NYC, or Boston, not even close to what Providence can offer. I don’t think New Haven has what your kid is looking for. New Haven’s main advantage is that one can get to Boston and NYC by train, about a 2 hr train ride to each. Same can be said for Princeton - a little over an hour to NYC by commuter rail.
You should look into Brown, in Providence. It’s a very strong school, in a decent city. It may be too small for your kid, and Providence may not be the exciting big city he wants, but Providence has undergone a lot of renewal, is a really great city to live in.
NYC schools. Columbia, which does have a campus, but is right smack in the urban upper west side. Barnard, if your kid is female, is a slightly less selective, somewhat more personal, woman-centric school but with access to all of Columbia’s resources. NYU, in Greenwich village, is in a very attractive area of the city. A less competitive alternative might be Fordham, in the Bronx. Things to do in NYC? You could tourist there for a month, and not run out of things to do and see. A ride on the Staten Island Ferry, to see the Statue of Liberty and the lower Manhattan skyline from the water. A walk through Central Park. A walk through the Village. SO many great museums. Broadway show, Times Square, list just goes on and on. Every cuisine in the world, at all hours of the day. Need I go on? Know that the subway has become an underground shelter for the homeless, and crime is rising - be careful. Honestly, this applies to all the cities I’ve mentioned.
One thing I’d keep in mind - your time is limited, and the primary purpose of this is evaluating schools, so do a TON of research ahead of time, and think of touristing as a palate cleanser in between schools. Take pictures in a way that allows you to add captions, to help you remember what you saw. For me, that would have meant Whatsapping them with a caption to the other parent who didn’t come on the trip, but it would have also helped me to remember which school had those horrible, tiny cinderblock-cell freshman dorms (are you listening, Northeastern?), or which school had the enormous enclosed area with the huge library steps, giving it a cloistered feel even though it was right in the middle of the city.
Remember that aquariums, zoos, and even non-specific-interest art museums are largely the same in many cities around the world. Unless your kid is planning on majoring in Art History, they are not the best way to get to know a city. Neither is a whale-watching trip, which, since the water stays cold much longer than the air, would probably be frigid misery in April - I’m not even sure if that’s when the whales are feeding off of Cape Cod - and would take a long time out of your day, and not help one bit to get a feel for a city. Same goes for sports, although if for some reason kid insists on taking in a ball game, the place to do it is Fenway, since it’s a fairly short walk from BU and Northeastern, is still part of getting a feel for a city. But again, I’d consider that not the best way to get familiar with a city on an East Coast swing college tour - the best way is walking, walking, walking.
Pack your walking shoes, and your raincoats and umbrellas, and layers. April in the Northeast can be anything from snowstorm to warm, late-spring beauty with tons of daffodils.