I highly recommend staying around Baixa/Chiado which is "down town" and is relatively clean and well presented - this area is historic and was built after the 1755 earth quake, though it's looks a little more modern than other areas around Lisbon which have a little more rustic charm.
Baixa/Chiado is very accessible and only 20 min from the airport by taxi (~€12-14). The area is well serviced by trams and in particular tram 28 which will take you to all the major tourist sights and is kind of the must do tram experience. The area is well serviced by restaurants (albeit touristy ones), a large supermarket and shopping is also very nearby (easy walking distance). There is also access to Hop-on, Hop-off tour bus services which run from the square nearby. The area is busy at night but not so much that it will keep you awake until all hours.
I would recommend renting an apartment if you're looking for good, cheap accommodation. On both occasions we've used this private company which is run by some Portuguese and German families. We stayed in the Santa Justa property and more recently another similar one in Santa Catarina. The one in Santa Catarina was huge, well appointed and easily accessible by Tram 28 but there was construction going on in the area which was a minor annoyance in the early morning.
Lisbon throughout is quite safe during the day and night though it has a slightly intimidating facade with quite a lot of derelict buildings about and lots of graffiti - but I personally like that about Lisbon. You can wander around most central districts pretty safely without worry and we've never had any trouble. Lisbon does appear to have a obvious drug problem - every day we were offered drugs in broad daylight and at night around night spots (bars/clubs). Having said this, they never bothered us past a slight mumble of "hash" (or whatever they're selling) and if you just ignore them or shoo them with a brief wave of your hand they wont bother you any further.
A few places I would recommend for lunch/dinner:
Restô do Chapitô (near castle) - has fantastic views over Lisbon and the bay. It is a functioning circus/talent school and if it is in they will have circus performances and performers around the tables, would be great for kids.
Senhora Mãe (near castle) - fantastic food, great service, nice ambiance.
A Travessa - difficult to find but the food is amazing and the setting is quite unique (in an old monastery) Maybe not appropriate child friendly place but still worth noting.
Portas do Sol (near castle) - Great for a drink and a snack, great views over the bay.
A great day trip with children would be to go to the beach out at Cascias. Take the train from Sao Paulo (a short walk from Baixa/Chiado) it takes about 30 min. Sintra is also worth a day trip, though I've unfortunately not been myself.