If you don't have much experience running wire, an easy option in some cases is to run the wire along the exterior of the house. You punch out the wire at the source, run along the siding, under the gutters, etc. and then drill through the wall to bring it into the house. This is easiest on exterior walls, but you can also bring the wire into the attic and then down into a wall.
If you choose this route, make sure to leave a drip loop and seal all of the holes with caulking or expanding foam. Use a low voltage rework box on the inside, and seal and holes you made in the vapor barrier with tuck tape.
Running wire inside is usually doable with minimal damage but requires a good plan and some experience with a flexible drill bit, fish tape, etc. Remember that at every location you choose to terminate a wire, you will need a cutout in the wall for the rework box. This is usually close to the ground, so you can easily drill down through the bottom plate to get between floors.
You can remove other wall plates for existing phone lines/cable and use cutout as an access point for running the wire. Similar but more involved would be removing existing electrical boxes in order to use the hole (make sure to shutoff the power!).
This combined with a few techniques for hiding wires such as behind baseboards or crown molding will usually allow you to plan a wiring route with minimal wall damage.
If you've never run or seen cables run before, you could get an electrician, alarm installer or other installer to do a few runs for you while you learn about how they do it.
Do yourself a favor too and pull multiple cables to each location. 2-3 Ethernet and 1 RG6 is a typical run that I have to multiple locations in my house. Conduit is nice but gets expensive to run and is not as easily installed as flexible cables alone.