Speed and climbing ability depend on you more than on the type of bike that you ride.

That said, I have an old, rigid-fork mountain bike that I built up as a hybrid, a road bike, and a cyclocross bike (road bike with wider, knobbly tires). I'm definitely faster on my road bike than on either of my two other bikes. The more aerodynamic position on the roadie and cyclocross bike is much more comfortable for my back (I prefer to stretch out), and the drop bars mean I have a variety of hand positions available for longer rides so my wrists won't hurt. The road bike is less effort to ride. It was a relatively inexpensive bike and doesn't quite fit right (it's close), but it's fairly light. I can't really climb hills very well, but that's more a function of ME and my conditioning. It certainly has climbing gears.

One thing you might notice is that climbing on a road bike might initially be harder. That's because hybrids are typically geared lower, so you can be in the same "numerical" gear (say, 1 up front, 4 in back) and it'll be harder on the road bike than on the hybrid. Again, that's mostly a matter of getting used to the bike.

Go to your LBS, tell them what you're looking for, and your budget. Test out a bunch. If you're looking to do longer rides, look at the more "relaxed" or "endurance" road bikes: Fuji's Sportif line (I hear they have a new one with disc brakes! Fancy!), Giant's Avail or Defy line, Cannondale's Synapse, and the Secteur/Dolce or Ruby/Tarmac from Specialized. The Jamis Satellite or Quest are also very nice bikes. Try both women's and unisex. But, don't be afraid to try the more "aggressive" bikes too.

Oh, and have fun! This isn't supposed to be a chore!