Two weeks in August with kids. Suggested itinerary?

Other people have asked similar questions, here' some of the threads:
https://www.lonelyplanet.com/thorntree/forums/europe-western-europe/topics/portugal-with-kids-this-easter
https://www.lonelyplanet.com/thorntree/forums/europe-western-europe/portugal/1-week-trip-to-portugal?page=1#post_19893936
https://www.lonelyplanet.com/thorntree/forums/europe-western-europe/portugal/pushchair-stroller-in-lisbon
http://onetinyleap.com/lisbon-baby-toddler-kids-top-5-activities/
http://www.golisbon.com/practical-lisbon/family-lisbon.html
Hope this gets you started.

Self and Gf spent a week in Lisbon in July 2015 and then did a second road trip later in the year, moving around every few days.

I have no kids, but GF has and she's now a grandmother too who helps out with the grandchildren.

Her view is that the first option of Lisbon and Cascais is very workable given that Cascais is a short rail journey away from Lisbon and you can also get to Estoril on foot/pram or to Belem and back to Lisbon on the train or to Sintra by bus.

Pavements in Lisbon not really pram/stroller friendly and the seven hills that Lisbon is built around could make for a lot of pushing of said prams/strollers! OK for a couple of days after the flight maybe...

Be prepared for high temperatures though and high uv levels too - we were using Factor 50 sunscreens in July last year and SPF clothing too. Bottled water is cheap though, just make sure that you get some small change as soon as you can as small shops and some cafes don't always appreciate even €5 notes for small purchases.

The Faro option may be worth considering, but the Algarve may well be crowded this year as it's a favoured destination among Europeans who aren't going to Tunisia or Shel-am-Sheik because of problems over there.

Train journey around 3 hours to Faro. Doable in hire car, but these may have stick shift rather than auto boxes. Make sure that there's a toll road device in the car and that children's car seats can be hired too.

And be sure to read this too!

https://www.lonelyplanet.com/thorntree/forums/europe-western-europe/topics/goldcar-my-review

As far as food shopping goes, mini-markets usually have good selections of meat, fruit and vegetables as do Pingo Doce and Lidl supermarkets if you are renting an appartment.

Enjoy Portugal!

With kids, I would recommend Cascais for the whole time. There are several nice, kid-friendly parks as well as beaches. I rented have rented apartments with kids that age in a variety of locations in the Lisbon area, and Cascais should have most of what you will be looking for. Note that the Portugues are particularly kid-friendly, and staying in the same place for two weeks will improve your chances of meeting local people through your children.

Thanks - all great tips!

Is there enough to do if we stayed in Cascais for two weeks? Is it feasible to go to Faro for a day trip? Or better to stay there for a night? What about Evora, Sintra, Estoril? All easy to access if Cascais is our base? If not we are considering 3 days in Lisbon, 7 days in Cascais, and another 4 days in Faro or nearby? But if we can avoid that and keep Cascais as our base that would be slightly better.

Evora is about a two hour train ride from Lisbon - 4 trains a day. Station is outside main centre - 20 min walk I seem to remember from our first road trip in 2013

There's a bus service between Cascais and Sintra - 30 to 40 minutes ride away

Estoril can be walked to from the centre of Cascais - we took a train from Lisbon last year, got off at Estoril and walked to Cascais. It's about 2km.

Faro is a three hour and twenty minute train ride away from Lisbon - 2-3 trains a day

We spent a few hours going around Faro Old Town and that included around 90 minutes having a leisurely lunch. We were staying in Tavira for a week in March this year and got the train to Faro.

I'd suggest getting hold of the Rough Guide and Lonely Planet guides to Portugal as in this case they may well complement each other and give you a better idea of what's what. If you're just going to be around the Lisbon area though then go for The Pocket Rough Guide To Lisbon as IMHO it's a lot better than the LP take on the Lisbon theme. Be advised though that the Toy Museum in Sintra closed down in 2014, but it is still mentioned in that Pocket Rough Guide.

May be worth staying in Cascais and travelling out using Viva Viagem travel cards - rechargeable travel cards that cover most, but not all, means of public transport (trains, trams, most buses and the Metro in Lisbon).

You did not mention why you wanted to go to Faro. The beaches in the Algarve are beautiful, but there are also nice beaches in Cascais and in other locations close to Lisbon. Although you can find a few towns in the Algarve that are still primarily Portuguese, I find that area to be unpleasantly "touristy".
Of course, Cascais also has it's touristy areas, but in my opinion it is still very much Portuguese. There are foriegn expats living there, and there are other non-Portuguese residents there that work for various multi-national businesses. This is a friendly community with good public transportation and a good environment for small children. As mentioned in an earlier post, there is a municipal bus you can take to Sintra, and a commuter train to downtown Lisbon (and many interesting points between). Considering the age of your children, I would recommend staying in one place for your entire visit.
In addition to the guide books mentioned in earlier posts, I would also recommend Rick Steve's guide for Lisbon.

You MUST go to COIMBRA. It's a magical city with the oldest university of Portugal. You have also the marvelous Botanical Gardens, The Downtown and another jewels of this tiny country. Once it had been the capital of Portugal (the first one after the expansion of the territory).
You should go also to SINTRA, SERRA DA ARRÁBIDA, MAFRA, ÓBIDOS, NAZARÉ and AZENHAS DO MARwhich are near of Cascais, and also GUIMARÃES,PORTO, GERÊS and DOURO VALLEY, which are in the North. If you have time, I recommend you to go to SERRA DA ESTRELA, VISEU, SERRA DE SÃO MACÁRIO, PIÓDÃO, FOZ D'ÉGUA, SORTELHA, BELMONTE and MONSANTO.

There is no real reason to split the trip between Lisbon and Cascais. For all practical purposes, Cascais is basically a suburb of Lisbon. Also, Monasterio de Jeronimos and Torre de Belem are on the train line between Lisbon and Cascais.

Algarve is a package tourist destination, especially so in the summer. With the political situation in Turkey, Egypt and Tunisia we can expect another record year this year, with tourist numbers easily hitting 8 figures (10+ million), and basically all of them arrive June-August. I mean, I don't want to judge, there is nothing wrong with wanting to be a part of that, but, you would have to pay me to do it.

Also, there is nothing to see or do in Faro. The fact that you picked Faro out of all the places in Algarve suggests you might wanna pick up a guidebook and have a flick through.

Is there enough to do for 2 weeks out of Cascais? Well, if you want to go on a day trip every day, then no. If not, probably it'll be OK. With small kids chances are you will need to split Sintra into 2 day trips. You'll probably want to visit Lisbon 2-3 times. Then there is Cabo da Roca, and there are plenty of beaches around. And you can drive to towns like Obidos, Peniche, Tomar, Batalha, Alcobaca, also Evora.

I encourage you to pick a nice place in the countryside, close to a beautiful little town, with a pool, and stay for 3-4 nights, somewhere in Alentejo or the Beiras.