6-7 nights in Portugal at the beginning of July

Hello,
We have a family of four and we traveled through Portugal last summer.
In Lisbon, we stayed at the Ibis Lisboa Liberdade. It was about 5 minutes to walk to the metro. We didn't eat breakfast at the hotel. Right down the street, was a cafe called A Padaria Portuguesa. We even walked to the center of town. Took the fun trams when needed. Don't forget to go to Belem while in Lisbon.
We did Sintra as a day trip.
Porto was a train trip with one night spent there. I would have liked s second night. We stayed at the Intercontinental Hotel, we had points for that.
Back in Lisbon, we rented a car from Hertz. We drove to Lagos and spent the day there. Drove to Seville. Spent 3 nights there. Then drove back to Lisbon, but stopped in Evora.
If you need any additional help don't hesitate to PM (personal message) me. Happy to help.
Have a super adventure!

If you only have 6-7 nights, I would encourage you to drop either Porto or the Algarve. Remember that you basically lose a day when you change hotels. So, trying to do 3-4 places in that amount of time will result in almost full time travel and no real time to see the places you are visiting. I do understand the desire to see as much as you can, but you could easily spend the whole time in Lisbon with day trips to Sintra and, maybe Evora. There are lots of beaches close to Lisbon, accessible by commuter train.

To my knowledge there is no train from the Algarve to Seville.

Drop the Algarve and visit the beaches near Lisbon. Accommodation has booked up very quickly this year in the Algarve and you may struggle to find somewhere just for 2 nights. You need longer than 1 night in Porto.

There are no trains between the Algarve and Seville, but there are several buses a day. Booking required.

There is very little cross-border trains service between the two countries. There are also no trains between Lisbon and Seville, again a bus, or a flight. (And the trains from Lisbon to Madrid or San Sebastian are also overnight only.)

Hi Lynn
I agree with Jennifer to focus on Lisbon, Sintra and beaches near Lisbon. I would consider staying at least a night in Sintra as there are lots of castles and gardens that the kids might enjoy for more than one day. It is also cooler. There is a small town very near Sintra called Sao Pedro that had a couple of excellent restaurants and was not touristy.
When we travel with our 3 kids we find apartments often our best bet, both for the space, price and ability to eat some meals in. I had good luck with VRBO.