You probably want to do some research regarding when to go. The March timeframe may be drier than June. There seems to be a wide variety of camping options as well as rustic, low budget accomodations depending on the exact island you target. I would say you may want to avoid Bora Bora, just based on the fact that it caters pretty exclusively to the well-off honeymooners based on what I saw.
Moorea may be the best place to start an adventure in the Society Islands. It is kind of the tourist anchor for the islands and has a very big (relatively) jungle-like interior. More waterfalls and mountains to explore.
Huahine is also supposed to be a cool island where surfers trek to in search of great waves. That usually means more relaxed, casual type accomodations will be there.
As a single woman, I think you will be quite safe, with maybe the possible exception of being in a bar in Tahiti's capital city. Does not sound like that is a worry for you though.
There is a local freighter that makes a round of the islands every week. I believe it carries passengers as well. I also think it makes it way to the Marqueses which are a more rural set of islands in the far north portion of the society chain. More mountaineous and less geared for the honeymooner.
By all means at least take a mask for looking under water with. The marine life and reef formations are phenominal! If you have ever considered scuba diving...now is the time! Even if you do not get certified before you go, you may want to look into taking a half day or full day type scuba trip, if they can give you a temp. cert for the day. You would not be disappointed.
You may also want to consider the Cook Islands. From what I gather, they at least match the Society Islands for beauty and probably hold a more authentic flavor as they are not quite as popular. I gather that Tahiti is getting more popular year after year.
And if you really are interested in treking and backpacking while on a tropical island...nothing compares to the Big Island of Hawaii, IMO. It has the Waipio Valley with some of the tallest waterfalls in the world and trails that go in and out around that....and then on the opposite side of the island is Volcano National Park which provides opportunities to see all kinds of nutty stuff!