In English if I didn’t hear someone, I would use any of the following: “What was that?” “Say it again” “What did you say?” “Sorry?” “What?”
What phrases are used most commonly in Portuguese? Would you say something different to a friend or family member versus a stranger?
I often want to avoid “Não entendo” or similar phrases because I want the speaker to say it again rather than just switching to English.
Hey, Molly. Well, you have a few different options. Personally, in a casual conversation, I often just say “O quê?”, either on its own or preceded by whatever part of the sentence I did grasp.
For example:
Person A: Tens uma caneta a mais?
Person B: Tenho o quê?
Person A: Uma caneta a mais.
“O quê?” can be a bit blunt, so you could also casually say “Diz?” or “Desculpa?”. “Podes repetir?” is also a good option. To make it formal, just change it to “Pode repetir?” or “Desculpe, pode repetir?”. “Não percebi” is also fine, but I can see how that might push people to switch to English.
Ahh the perks of having all the Learning Studio recordings at your fingertips 
♂ Here are a few audio recordings for some phrases discussed, in case it helps:
Pode repetir? (Can you [formal] repeat?)
You can add a “por favor” on the end for extra manners:
“Não entendo” (I don’t understand)
@Joseph, would you say that “como?” is also a gentle and versatile way of asking someone to repeat themselves in most situations?
Sure! I forgot about that one, but it’s also a good option 
@joelrendall acho que “Não entendo” é o que a Molly está a tentar evitar. “Não entendo” realmente abre caminho ao recetor a mudar para inglês, porque parece que a pessoa não entende o que foi dito.
Eu acho que “Desculpe(a), não ouvi” , “Como?” e “Perdão?” são boas alternativas.
Por outro lado, o curto “Diz?” pode funcionar muito bem entre amigos. Mas, cuidado porque em situações formais o equivalente “Diga?” pode soar um pouco agressivo.
Em relação às hipóteses já mencionadas, acho que “Pode repetir?” soa muito formal e “estrangeiro”, e mais uma vez pode levar à mudança para o inglês. Em relação ao “O quê?”, acho que este deve ser reservado para situações muito informais (tal como o @Joseph disse e bem).
Is it ok to say Novamente por favour? That’s what I usually say but wonder if it is correct.
It doesn’t sound natural to me, but it’s not wrong and people would get the point, I think 
I’d still point you towards the other options we’ve covered higher up.