I have not looked at that private-transportation info in the Spain guidebook, so I don't know how it is phrased, but 60 euros seems cheap. I bet that's one way. And round-trip might be more than double, depending on how long you need for the driver to wait. ViaMichelin.com gives the distance as about 24.5 miles and estimates the driving time at 40 minutes.
Time-wise, this sounds doable if you get an early start. Cost-wise, I don't know what you're going to run into. I'd be inclined to schedule the Port Lligat stop first, because if you arrive late at your second stop, I suspect you'll find the Dali Theatre-Museum more flexible about still letting you in outside your ticket entry-time window. (No guarantees.)
There's a bit of a complication in Figueres with respect to the train stations (note the plural). The fast trains go to Figueres Vilafant station, about 1 mile west of the Dali Theatre-Museum. There is bus service timed to the AVE schedule that will get you to what I took to be the center of town. I think there's a stop along the southern edge of the large green area you can see on the Google map. There may be a stop closer to the Theatre-Museum; I'm nearly certain that bus continues on to the regular Figueres train station, about 0.6 miles southeast of the Theatre-Museum.
However, if the AVE is more than a few minutes late, in my experience the bus doesn't wait and you'll have to hope there's a taxi still hanging around (don't stop to use the station toilet!), or you'll either have to walk or call a taxi. I think there's usually a taxi telephone number posted outside train stations, but I've never tried to summon a taxi that way. Figueres Vilafant isn't in a busy area, so there won't be random taxis cruising by.
The buses to Cadaques depart from right outside the regular Figueres station, or at least they did in 2016. That seemed a busier area.
I spent a few nights in Figueres in 2019 during the May flower festival in Girona, when Girona's hotel rates were unusually high. I walked around Figueres a bit and didn't find it particularly interesting compared to dozens of other places I've seen in Spain. I don't think you'll be missing much if you just see the Dali Theatre-Museum (and the jewelry collection if you have time) and then move on.
I have a suggestion: Consider spending your first two nights in Girona and relocating to a Barcelona hotel later in the day on your second full day in Spain. Doing so would loosen up the schedule a bit on the day you go to the two Dali sights (your first full day in Spain). If you can keep moving on your arrival day, you'd be able to see a bit of Girona before crashing. You could see more of Girona on your second full day before heading down to Barcelona.
The fast trains between Barcelona and Girona take only 38 minutes but require seat reservations and usually ticket purchase well ahead of time to avoid jacked-up last-minute fares. But there are also Media Distancia trains that take 70 to 80 minutes, which wouldn't be disastrous if you weren't rushing to hit two museums on the travel day. So you could wait until you arrive at the Barcelona airport to decide which MD train to take. You could even do that on the day you shift to your Barcelona hotel if you want to make a spur-of-the-moment decision about how much time to spend in Girona.
Alternatively, you could spend those first two nights in Figueres (normally with much lower hotel rates), but as I have said, I find it not all that interesting a city, whereas Girona is a gem.
