The nearest three hospitals to Bourne would be Falmouth Hospital, Jordan Hospital and Toby Hospital. All are about 15-20 minutes away. There is also a clinic right in Bourne at the 'five corners' near the Bourne rotary. I have been there a few times and they have always been very good.
I personally see Dr. Dinn, a dentist in Sandwich, MA but many of my friends go to Bourne Dental Associates and have great things to say about them.
Please let me know if you still have questions - I know this was posted quite a while back :-).