Go big or go home. If you're going through all the trouble to pull wires through existing walls, don't pull just one. It's only slightly more work to pull two wires than to pull one. Doing the project a second time in a few years is twice as much work, though!
Think about everything you might want (IP networking, surveillance cameras, cable or satellite TV, telephone, speaker wires, home automation control signalling). Then add a bunch of extras. Cat-6 is a good choice, because it can carry most things you might want.
Consider a single conduit. During a major bathroom remodel & replumbing job, I had the plumber put conduit from the attic to the "wiring closet". He went a little nuts, putting two 4" conduits in - one for high voltage, one for low voltage. This makes it relatively easy to add a new service anywhere. Unfortunately, this was after I had finished running structured wiring.
Since a small part of your basement is unfinished, you might want to run conduit from there to your attic, and make that your "wiring closet". Then you can pull from each outlet to the attic, and easily run it down the conduit.
Star topology. Pull everything to a central location, and patch them together there. I used the space under the stairs.
How to pull up to an attic: In your attic, drill a large hole in the sill plate. It should be big enough to fit all the wires you want through it, but small enough to keep the strength of the sill plate. Avoid the wall studs below.
Cut a hole in the wall below, for the outlet.
Push a fish tape down from the attic. Have a helper watch the hole for the tape, and grab it with a bent coat hanger. They attach the wire bundle, and you pull it back up.
To push/pull from attic to basement, you can cut a hole on the floor in between, and do it in two steps. Or you can fish from both ends at the same time, using fish tapes that have hooks on the end. When they hook each other, pull the attic tape up to retrieve the basement tape. In the attic, tie on the wire bundle, then pull it down to the basement.
To pull between finished floors you need to cut a hole in the wall to drill a hole in the sill plate & fish between the floors.
See also my answer on how to pull wire to a basement or crawlspace: How do I run ethernet, speaker wire, and coax through a wall into a crawlspace?
The more holes you cut, the easier it is to pull wires, and the more fixup work you have to do.
CAUTION ALERT ! Dealing with a failing plaster and lathe ceiling can be very tricky. Before you make the decision to tear it down, which as Chris mentioned, a messy job, or simply put drywall over the existing plaster, you need to carefully evaluate the soundness of the lathing to the strapping or joists. What I mean by that is, are the lathes still firmly attached to the joists and the problem is the plaster "keys" failing from around the lathes, or are the lathes themselves falling away from the joists?
With that said, covering the old failing plaster with drywall is fine. If the lathes are solid to the joists, then putting drywall directly over the old and using screws long enough to reach the strapping or joists will work fine. However, if the lathing itself is failing, then you must install strapping 16 inches on center, perpendicular to the joists, securely screwed to or ring nailed to the joists. The purpose of strapping is twofold. First reason is to support the failing ceiling securely to the joists. The second reason is that if you just screw drywall to the failing plaster and lathe, the weight of the ceiling will be against the drywall and could pull the screws through the drywall board allowing it to sag or joints to crack now or in the future.
I must say that even though tearing down the old ceilings is a big job, it certainly gives you a clean start and an open view to correct any old wiring problems, add new lighting circuits, install new flush mounted electrical boxes, and install new level strapping etc.
Best Answer
My house was built in the 1890's and is predominantly plaster & lathe. I've mounted everything from small pictures to 10' long by 8' tall shelving to the wall by simply running drywall screws into the wall. I've tried using a stud finder to find studs, but the lathe befuddles them. I simply drive a screw wherever I want to hang something and have never had a problem.
When mounting on plaster and lathe, I'd recommend simply driving screws through the mounting holes and into whatever they find to hit behind them. As small and light as these are, even if all 4 screws only find plaster, they'll still hold just fine.