I total agree with Katarina about Matosinhos is the best option to live special with kids... you can enjoy the beautiful city park of Porto and the shoreline to walk, run, cycle, etc....
Vila Nova de Gaia can be nice if you don't need to go to Porto, because the traffic to Porto is really bad...
Maybe you already have this covered, but for staying for longer than 3 months you need a certificate of residency (or something of the sort, depending on the city hall the name changes sometimes). In Vila Nova de Gaia it covers you for the duration of the work contract, and I think in Porto it's the same but not sure. After 5 years or something you can get a permanent certificate. If the contract works with automatic renewals, you might need to get one each renewal before you can get the permanent one. It has a symbolic cost of 5 euros I think.
If you don't get one, you probably get scot free anyway, but you might get caught, and in that case, saying you didn't know won't be any help and you will be fined.
If the requirements are the same as in Vila Nova de Gaia, you need proof that you live in Porto (a bill letter would be enough), a copy of your work contract, pay receipts, your id or passport and you need to fill a form. Again, these requirements are likely similar but vary slightly between city halls, so you might need to provide different documents.
Source: went through this with my gf about a year and a half ago. And again six months ago.
Martina is right about everything. From my point of view Villa Nova de Gaia is the best place to live, Matosinhos is good but you have to be ready for a strong wind from the Atlantic ocean. There is a Dritish school (in case you have kids) at the area of the Douro's estuary , so maybe this area together with Matosinhos (as it's very close) are the best places for yuo to settle down.
The climate in Porto is usually compared with London's climate so I have an idea that it won't be a problem for you.
the areas really depens on what you like and what you want to do
if you plan to work in Porto from the early morning on, i would not recommend Vila Nova de Gaia, due to the traffic (only with a few exception).
if you value the beach, then the coast of Vila Nova de Gaia (south of Porto), Matosinhos (north, but really connected to Porto) or even Foz (the most expensive neighbourhood of Porto) are the best areas.
if you value parks, then the Boavista area, near to the city park would be the best.
if you value night-life, then downtown (Baixa) or Ribeira are the best.
if you have any trouble regarding renting/buying, just let me know and i'll see if I can help