It sounds as if you have experience traveling to Europe with your kids, but I'm amazed that your experience has been that all of your family of four is capable of doing real sightseeing on the arrival day. I'm totally zombified due to jetlag and sleep deprivation. I try to do something outdoors that keeps me moving; otherwise, I would fall asleep. I would never commit to doing something that requires concentration. So I think the idea of walking around the Barri Gotic or the waterfront would be great.
I don't find the Ramblas very interesting, and it is mobbed (including pickpockets), so I don't think it's a good destination when you may not have your wits about you. If you find yourself there because it's on the way from your lodgings to wherever you're headed, that's fine; it's just not worth a special trip unless you're headed to La Boqueria, the market, which isn't a bad (though not cheap) place to satisfy the munchies at off hours.
I love the modernista sights, but they are quite expensive--at least for adults; I don't know about the entry fees for children. You'd need to buy the tickets ahead of time, else you probably wouldn't get in, so you'd be betting a good bit of money on your collective ability to keep moving purposefully when you may not all feel like it. That's not something I'd recommend for Day 1.
These are the six sights Barcelona sights for which you need to buy tickets in advance if you're interested in going inside--assuming you're not willing to risk spending an hour or more in line and maybe find out that the day's tickets are already sold out: La Sagrada Familia, Parc Guell, Casa Mila/La Pedrera, Casa Batllo, Palau de la Musica Catalana and the Picasso Museum. You can spend extra money at some sights to buy a wildcard ticket that doesn't lock you into a specific entry time, but as I said, those places are expensive to begin with. If you buy the ordinary, timed tickets you have the fun of trying to guess how much time you'll need to see your first destination and get yourselves from there to the second destination (and maybe eat a meal along the way). It is not a simple task to develop a sightseeing strategy for Barcelona!
As I understand it, you have the same situation with the Dali Theatre and Museum in Figueres, so I'm afraid you'll need to decide ahead of time whether you want to see it and buy the tickets in advance if you do. I don't have kids, so I'm not in a good position to comment on what yours might like or which potential day-trip would be best. But you have two train options for Figueres, a fast train taking 55 min. and a slower train taking about twice as long. Last-minute tickets on the fast train will probably be rather expensive (at least $40 roundtrip for adults).
It is highly recommended that you slot one of the book-ahead sights (if you want to see some of them) first thing in the morning each day. You have three days, so pick your top three if you have more than that. I think Parc Guell suffers a bit less from being crowded (it's outdoors, so it's easier to spread out), so that wouldn't be my top priority for one of the first-up slots. You're traveling in April, so it doesn't seem that heat would be too big an issue if you end up deciding to go there at mid-day.
The hospital (Sant Pau modernista site) has so far not been so crowded that advance tickets are required. That could obviously change at any time, as word about it gets around. But with luck it will still be a just-walk-up option when you get to Barcelona. It is large; folks really interested in modernista architecture may need more than two hours there.
Oh, one thing that the kids might get a kick out of: There's a spot in the Barri Gotic where you can walk through a door and find three or four Roman columns inside the building. It's really cool. I'm nearly certain Rick mentions this in his book. I think that door isn't open super early in the morning.
