Hi Varsha,
In South Goa you will get plenty of budget stays. Palolem beach side is definetly one safe place I recommend. You can also stay at Pollem and Colva. You get good stays starting at 1000 INR.
Hope this helps.
Hi Picking up some good hostel will be good idea for both budget as well as comfortable stay. You may check some of the best hostel on below link in Goa https://prf.hn/l/reNlX8E Thank You
Cheapest option would be an Airbnb or OYO if you have anyone to split the price with. If solo best would be a hostel. Good hostel options you'll get just Google.
Would prefer to stay at palolem, there are tons of beach front stays there are reasonable price. Dcosta, Brendon, presleys fernandes are some good ones. Just simply walk down the beach and hop around stays and bargain over there itself for best price. Try to get a place on the north end of the beach rather than south one because it's less crowded and has near access to sundowner point and backwaters.