There's a Bus that goes to Mapusa that starts in Sinquerium and takes you to Candolim,Calangute and then to Mapusa... There also a bus to Panjim that can take you from Calangute to Candolim then it goes to Panjim... A seperate bus can take you from Calangute to baga I think (never used).
Bus to Palolim - first you need to get to Panjim Bus station (9 /- Rs), then Panjim to Margao (around 16/- Rs/-) - This is a guarenteed seat service non stop. You then get another bus from Margao to Palolim, it's quit a long journery but it's cheap.... it's at least worth getting to Margao by bus as the panjim margao route is very quick...
Not sure about Ashwem - I guess you have to go via Mapusa..