The yellow cables are probably Cat5e, which has 4 pairs of wires. (The beige cables are probably Cat 3, which might only have 2 pairs of wire). A single phone line only requires 1 pair, so as you can see the other pairs are just twisted off.

You definitely can reuse the cable for ethernet access, but you will lose the phone jack obviously.

Here are some things you should know about working with ethernet cabling:

• Unlike phones with are relatively low-bandwidth, ethernet cables need to be in good condition with the pairs twisted around each other properly and with properly terminated connections. The mess you show is fine for phones but will cause lots of interference with ethernet data. If you want to reuse the cabling for data you need to cut back the cable all the way to the yellow sheathing and start over with new CAT 5e terminations. The proper way to do it is to terminate all the cables at a patch panel, but you could also just crimp a regular RJ-45 on the end and connect them that way. You cannot just twist the wires together and expect to get a good signal. Also, at the other end of the cable (in the room with the jacks), you will need to do the same thing: cut back the cable to where it's clean, and start over with new RJ-45 jacks.
• Also unlike phones (and household electricity), you cannot split a cable into 2 cables just by joining the same color wires. If you have more than two devices you want to connect, you need to use an ethernet switch, which is a small appliance that will properly route the data packets between 2 or more devices.

(Lots of) Home runs are good

You are correct that you want to run a cable to each room from the central switch. In fact, I would run at least 1 more cable than you think you will need to each room, and consider running a line or 2 to other rooms as well - especially if your walls are open. Cable is cheap, and pulling 4 cables instead of 3 is no more work when done at the same time. If you decide later that you want a 4th jack, you either need a small switch (which does limit bandwidth, not really increase latency) or you need to open walls again to pull that 4th cable.

Use a patch panel

Rather than run the cable from the big switch to each room, you should have a patch panel in between. Patch panels basically change the type of connection on the cable (the back is a 110 punch down block, front is an RJ-45 jack), and are a simple pass-through.

This is to ease installation. Pulling cable through walls is best done when the cable is un-terminated. Terminating the cable (i.e., putting the RJ-45 jacks on the end) can and is done, but punching the cable down into a patch panel is so much easier, especially for someone who has never done it before (and it sounds like neither you nor your electrician has). The cost is marginal (again, go bigger than you think you need now), but you save on headaches during installation.

You would then get keystone jacks that allow you to punch down the cable on the other end:

You shove these into wallplates on an electrical box or low-voltage plate:

They make wall-plates with different numbers of openings (usually 1-6), so you can get what you need for each room.

Finally, you would need short (1-2 ft) "patch" cables to connect the patch panel to the big switch. Buy these cables pre-made, as you won't be able to make your own for less. These are typically stranded cable, as it's more flexible.

Your final setup would look something like this:

(the top-most device with jacks is the patch panel, the middle on is the switch, and the bottom would be your router)

Buy solid copper UTP (unshielded twisted pair) cat5e or cat6 cable, rated properly (usually CMR for typical in-wall installation, but you'll need Plenum if you plan to run it in HVAC ducts), and buy multiple boxes if possible. Standard is 1000 ft but smaller lengths are available, and they come in all different colors. A decent-sized house could take 2000-3000 ft of cabling or more, depending on how many runs and where the network closet is. Again, the more boxes you have, the easier installation will be (you typically pull 1 from each box at the same time, so if you want 4 runs to a single location, having 4 boxes is easiest).

If you want things a little cleaner, you can get a wall-mounted mini rack as well:

Just make sure to get one that has the depth and vertical space (measured in "U") you need. They also make ones with hinges that make patch panel installation a bit easier.

Most product images taken from monoprice.com