All the advice above is good (esp. watching Shug's videos).
I bought my first hammock going on 10 years ago (a Hennesy Ultralite Backpacker), and while I like it very much and have used it often in places I could never have pitched a tent, I find it far too cramped nowadays. Whether that's because I'm no longer a slender 65 KG or because I'm somewhat more precious about comfort at the end of a long day I don't know - but whatever you do, make sure the hammock is sized appropriately for you. Gathered-end asyms are unforgiving if they're too small.
The other thing that irritates me recently is the 'birth-canal' entry, as it was called; it's fiddly and irritating in the middle of the night, and I think the zippered option (which simply wasn't available when I got mine, even as an aftermarket mod 2QZQ style) would be vastly more convenient - though I haven't tried it, and I imagine a good hang is quite important so as to ensure enough slack is present to zip cleanly.
My next hammock will most likely be a WBBB DL 1.1, with a proper underquilt setup and possibly a 'gear hammock' below, though I do like the look of simonm's gear (linked above).
As far as tarps go, I've always found the tiny stock Hennessy diamond tarp perfectly adequate for staying dry, through some pretty wet weather (though nothing torrential); but I do love the
Warbonnet Superfly. Overkill? Maybe, but I like to go out in wet weather, and it's nice to have a refuge big enough for more than just sleeping.
Another point to consider is what other outdoor activities interest you. If I'm walking, my shelter depends on where I'm going. On the mountain bike, I usually take a tent; if I'm kayaking, I almost always take the hammock, as it can be almost impossible to find flat, dry sites in some places.
Good luck with your choice.