Yes, there are shuttle buses that can take you elsewhere should you decide not to stay in Cairns. I have used Sun Palm Transport to go between Cairns Airport and Port Douglas and there are other companies as well.

There are plenty of boats in both Cairns and Port Douglas which offer snorkel and dive tours to the Outer Reef. I'm a big fan of Wavelength (gone out with them 5 times) because they are a snorkel-only boat which takes a maximum of 30 passengers. They are based in Port Douglas, but you can arrange for a bus transfer if you are in Cairns.

Nywoman,
Tell us a little more about your interests so we can respond meaningfully.

Nywoman, I wouldn't visit 3 cities. This is not why people visit Australia. For the last leg I would consider going to Kakadu or Uluru (both in the Northern Territory) or Tasmania perhaps if you are interested in wildlife and walking.

Yes, some background information on likes/dislikes would be helpful but then this thread would suggest that Nywoman is fairly adventurous -

http://www.fodors.com/community/asia/nywoman-in-thailand-and-myanmar.cfm

For Cairns - although some would disagree - I would stay in Palm Cove or Port Douglas, a bit to the north. Cairns itself has (in my opinion) very little to recommend it. Most of the reef trips depart from Cairns, although some leave from Port Douglas. Wherever you stay in Palm Cove or Port Douglas, the hotel will be able to arrange a snorkeling trip to the reef for you, including bus pick-up and drop-off from wherever you're staying. They can also arrange the bus transportation to/from Cairns airport for you.

In Palm Cove, the beach is (literally) across the (fairly narrow and quiet) main road from all the hotels etc. Because of its location, most of the traffic is either arriving or leaving, i.e., really no thru traffic in Palm Cove, as it's off the main highway north to Port Douglas. Port Douglas has more shops etc. etc. but depending on where you stay, it may be a hike to Four Mile Beach. We've always stayed in Palm Cove (for what that's worth) and usually without a car.

Most tourists probably do the Kuranda trip - take the train up and the Skyrail back. Again, whereever you're staying can arrange it for you, including bus pick-up/drop-off from your hotel.

http://www.kuranda.org

The other main place to go in the Cairns area would be the Atherton Tablelands -

http://www.athertontablelands.com.au

Palm Cove is about 20+ minutes north of Cairns, Port Douglas about an hour or so. We've only been to Port Douglas when we had a car. You can certainly get a (regular) bus from Palm Cove to Cairns though I'm not sure how easy it is to get to Port Douglas without your own transport.

The video on this site will give you some idea of Palm Cove -

http://www.palmcoveaustralia.com