Simple=good.
Unless you will be dealing with large hills, Single Speed would be a really good place to start, IMO.

Also, just to avoid confusion Singlespeed (SS) is a single cog bike (one speed) with a freewheel, while a Fixed Gear (FG) lacks the freewheel. SS is totally workable and represents a really nice option to start with, and offers compatability to convert to an internally-geared hub at a later date, if desired. As mentioned by others, I'd avoid a fixed gear (no freewheel, no coasting) if you are getting started.

Since you express an interest in venturing off road, I'd recommend a SS mountain bike or cyclocross bike, either of which can be fitted with slick tires for your street use. By your post, it seems that your offroad needs might be met with a cyclocross bike or a "hybrid" that allows you to mount tires 32-40mm (or larger)
Look for frames that allow you to mount fenders and/or a rear rack.

The Torker Graduate might be an option. 5 speed internal hub (simpler, cleaner drive train, trouble free shifting), ready for fenders/rack, and sufficient for dirt roads, smooth trails.

Here is a good bang for the buck bike from Bikesdirect, which would necessitate some assembly/fine tuning, but would be a solid value.

ANother option from BD: http://bikesdirect.com/products/mercier/kilott_wt5.htm Sort of a drop bar equivalent of the torker graduate.