I've been hammock camping for about four years, and there are a few issues you should be aware of.
First, as already noted in the comments under your original question, insulation is critically important. I know that below about 65 F (18 C), I sleep uncomfortably cold. This is because your insulation (sleeping bag) beneath you is compressed by your body weight and the air under your hammock is free to circulate. Combined, this will quickly sap your body heat.
The Hennessy Hammock I use has an insulation package available as an option. This consists of an open-cell foam pad and an "all season" outer liner. If this isn't warm enough, you can add a Mylar emergency blanket between the foam pad and the liner. Most importantly, this system installs on the OUTSIDE of your hammock. Using this setup, I have camped comfortably at temperatures below freezing. Videos of this system and how it is installed are available through their web site at http://hennessyhammock.com .
Other options include using a foam pad or a Therm-a-rest style inflatable inside the hammock. Some hammocks have an option to install these trapped between two layers of fabric.
Second consideration is the way you pitch your tarp. In general, it should be perpendicular to the wind (lest it act as a funnel). If possible, the ridge line should run above the tarp (rather than under) since this configuration will not allow water to run beneath the tarp if it rains.
In addition to staking the corners of your tarp, make sure you keep the edges taut as well. If the wind picks up, the edges can start whipping around, creating a lot of noise at night. I run lines from the center of each edge to the tie-downs in each corner (which results in something of an 'W' shape) and pull these tight with a taut line hitch.
Finally, make sure you know your knots. If they slip during the night, you will find yourself dragging on the ground or with a tarp flapping in the wind. If the knots supporting your hammock slip, the dip in the hammock may lead to your head and/or feet being uncomfortably elevated.
Personally, I have taken to using hammock straps (such as the ENO Atlas) and carabiners with my Henessy. Once these are set and stretched, there is no noticeable sag during the night.