I would encourage you to get a good guide book - RS, and/or one more - to help you decide where you want to go. In a month you can really do the country justice, but I agree with not moving around too much. My 'scouting' trip before moving here was Porto, a week on the Silver Coast and Lisbon. I had a car while on the Silver Coast and visited a lot of places in central Portugal. I didn't do the Algarve until after moving here, but in a month, you can definitely see it, too. My favorite city to stay in the Algarve is Lagos. But, be prepared for big crowds in June, both in the Algarve and Lisbon. Last summer, Lisbon was reportedly more crowded than its ever been.
Don't try to drive in Porto or Lisbon but you really need a car in the smaller places and the Algarve. The trains between Porto and Lisbon then Lisbon and the Algarve work well and are very inexpensive, especially if you book ahead.
Airbnb apartments (and other vacation rentals) are plentiful and inexpensive. Especially in the big cities, air conditioning would likely be a good idea. The markets in Portugal are great with great seafood, fruits and vegetables. Lisbon has many, as do the Algarve and Porto.