There are many beaches in goa long way up to the north which are very beautiful, peaceful and less crowded. For example: Morjim Beach, Mandrem Beach, Arambol Beach.
You can also stay at the beach side shacks at Mandrem beach which are very affordable.
Being from North Goa, I can tell u that South Goa is definitely more peaceful and not over crowded with tourists. We too escape to the South for a relaxing weekend sometimes. Palolem, Agonda, Benaulim, Sernabatim, Cavelossim, Colva beaches are some of the popular beaches down South. My favourite is Palolem.
If you are planning to have a peaceful and quiet vacation in Goa, then you definitely have to avoid north Goa, and go to the South, where nobody disturbs the flow of your thoughts and nobody judges you if you stay still staring into nothingness for hours at a time. When I visited Goa last year, we stayed at Sea Front Beach Huts, at Patnem Beach, and was totally in love with that place. You can check this place, or browse for others, there are many. Will be glad if I can help you with further information. Do let me know.
The best thing about Goa is that it has the best of both worlds.
Partying and night life along with cool and calm vacation with your loved ones.
I'd suggest North goa for an adrenalin rush at hot an happening beaches of the north. e.g. Candolim, Baga, Anjuna. However, the south of goa offers serenity, peace and soothing environment where you could just lay back and enjoy your time peacefully. e.g. Palolem, Cola, Agonda.
This one definitely has to be South Goa. If you want a peaceful vacation, I guess, you would prefer to avoid the bustle and would love to stay in close proximity to nature. Cola beach in south is one such place that one can look forward to.
Yoo can switch to different places as per your choice in North Goa. Whereas in South Goa, you'll find cleaner beaches (cleaner than Baga and Calangute beaches. Otherwise, Candolim, Vagator, Morjim, Arambol in north are much cleaner), pretty sunsets and peaceful holidays but you'll miss the party.