First trip to Japan in June with family of 3. I need advice on what areas of Tokyo and Kyoto (6 nights each) to look for lodging. Looking for Airbnbs or hotel rooms with kitchen and laundry. Thanks!

For Tokyo – IMO, it depends on what you want to see and experience. I’d recommend someplace near a train / subway station that provides reasonable access to your priorities.

For Kyoto – I’m glad I stayed near the Kawaramachi station, close to Gion and Pontocho; convenient to the things I wanted to see and experience and lots of restaurants in the area.

Japan is very safe, you will need to master the public transportation anyway so it's just a matter of your preference. Not sure about your budget, but most moderate chains have washer and microwave for all hotel users, and convenience stores have vast varieties of to go food, or to be microwaved food which you can heat up in the store or bring it back to hotel. Some hotels allow you to use the breakfast area for eating in. Dormy Inns with various locations are one of them (in house free onsen) but not sure if they have triple room. Or try Asakusa Tobu where they have triple.

In Tokyo I stayed at the Asakusa Tobu Hotel. It was a really great hotel. Subway station across the road. Very clean and I liked the area... But it was at least a 20 minute train ride to a lot of things we wanted to see. I found that the train/subway system in Tokyo quite exhausting to get around.
I am planning to visit again, and I think I would prefer the Ginza area. But wherever you do stay make sure it is close to train/subway station.

In Kyoto, I stayed at the Hotel Mume. I liked the hotel and the area, but I think you might do better staying near a station.