I'm from Cascais so it's heartwarming to see the love for my hometown!
I suggest other places in what we call "The Linha" (the coastal strip between Lisbon's city centre and Cascais).
Estoril is lovely.
In Lisbon itself, my favourite places are:
Principe Real - Beautiful architecture and gardens
Lapa - some of the most affluent locals live in this neighbourhood
Belém - Centro Cultural de Belém/Padrão dos Descobrimentos / Mosteiro dos Jerónimos and, of course, Pastéis de Belém (they're are absolutely NOT overrated, nothing compares. They will ruin you for any other examples of this pastry)
Bairro Alto - quaint, typical quarter, but full of hip clubs, bars, artsy type shops.
Alfama - as above, more touristy less alternative
Castelo de São Jorge - take the tram up, walk the rest of the very steep hill and you'll get the nicest view of Lisbon. Lots of lovely cocktails and perfect coffee.
Docas - the docks in Lisbon are jammed packed with amazing restaurant and bars. The food is lovely and the view is super.
Tróia - an island across the Tejo (Tagus) river. Loads of dolphins and beautiful water/beach
I was born and raised in Lisbon, but haven't lived there in a long time. It's a beautiful city.
Hope you enjoy it!