As a first timer i would do it by taxi and stay overnight in a beach hut at Palolim. We have done it on scooters but stopped a few places on the way.

I agree with Paul ,for your first visit get a taxi and spend the night in Palolim, or you can book a trip through a local agent near where your staying where they will collect you and take you there,wait and bring you back next day.
From the extreme North Goa to Extreme South Goa is around 80 miles i think.

Nightnurse

When we did it on a scooter we stopped off in cavalosim overnight then went on to Palolim. We did it by taxi in February, I think it was £12 to get to Palolim from Calangute and £10 a night for a good hut near the beach and £12 back. You can get your driver to wait overnight for you but if you only pay for a single journey you have the option of staying longer or going to another resort down south. We liked Belolim. Did not like Colva.

Nightnurse

You can get an overnight trip at Palolim for £19 pp available at http://www.johnsboattours.com in Candolim. I think Daytripper don't do Palolim any more but could be wrong