How to keep an airplane stable on the runway in Kerbal Space Program?

I've found that all my runway wiggling planes are caused by the wheels. This happens in two instances; initial placement and plane body deformation.

Wheels need to be touching the ground exactly as they were intended to, you cannot angle them in any direction other then their intended direction when placing them. So the ones that go straight up and down can only be straight up and down, no angling left, right, backwards or forwards such as when you place the little wheel on the angled body of the cockpit part. The ones that are at an angle must be placed at that exact angle, you can't angle them to make them wider.

Plane body deformations can cause the wheels to touch the ground at a slightly different angle. Body deformation can be caused by heavy parts, many parts connected together, or forces on the plane.

From your description of your plane I would check the following:

• Center of lift is behind center of mass, and slightly raised
  When your plane speeds up, it sounds like your lift could be pushing your nose into the runway causing deformation or an oscillation with the same effect.
• The two engines have been mounted every imaginable way with the same result: Over the wings, on the fuselage, on girder segments, on the wingtips, etc.
  Assuming nothing is blocking the exhaust of the engines, the relation of your thrust to your center of mass could also cause the same problems as above.
• Wheel base is wide
  If your wheels are far apart there may be deformation caused by heavy parts or many parts between them, I would try attaching struts from each wheel to a single fuselage part. Also ensure that the wheels are placed "correctly".

You seem to have done a lot of things right. I'm finding it quite surprising you didn't manage to get the plane to fly yet. I haven't encountered oscillations like that in an airplane but I have encountered them aplenty in case of lightweight probes that have too strong control authority - too good RCS or reaction wheels, so I suspect the source here is the same. Try reducing control authority (or even generally amount) of control surfaces - rudder, ailerons, the front wheel (maybe even make it a fixed one).

Additionally if you have any off-center fuel tanks, double check the fuel is drawn from both at the same rate (same priority). Plane veering off course is not uncommon but uneven fuel draw can exacerbate it a lot.

Also hopefully you're not using any mods that use throttle (Throttle Controlled Avionics etc) as with engines as sluggish to react as jet, this is bound to cause trouble.

If that all fails, you may try building your plane lighter and with mightier (or just more) engines, so it just doesn't have the time to start misbehaving on the runway before you bring it up in the air.

one last thing to verify: look from the side, observe where your center of mass is, then how the control surfaces are located relative to it and if the way they react to your controls makes sense (just try to pull up, down, or roll, on the runway without even starting the engine or disengaging brakes and watch which way the control surfaces move). Control surfaces near center of mass sometimes get "confused" as to their role and do stupid things. You may need to change control authority to a negative value if they bend "the wrong way".

ps. Start with simplest, easiest planes you can and then increase complexity observing and mitigating problems as they arise. Starting from something with MK3 hull without experience in MK1 will be an uphill battle.

ps2. It's unlikely you're doing this but just in case: while on the ground use only yaw controls to change direction, don't touch the roll controls. Roll is the main way to change direction in flight, and best way to crash the plane on the ground.

I haven't found a sure-fire solution to this, but I think I know what causes it. If you have more than one engine, you can get asymmetric thrust at low speeds. Thrust is determined partly by velocity. If you (or SAS) turns even just a little, the engine opposite your turn will have a higher velocity than the one towards your turn. At low speeds, this difference can be significant. This makes one engine generate more thrust than the other, exacerbating the turn. SAS sees this and turns the other way, which just causes the same problem in the other direction. Eventually you either crash, or you reach a high enough speed that SAS wiggling can't cause a significant enough difference in engine speeds.

In addition to the tips others have suggested, try locking steering on all your wheels. This should reduce ground control authority and make oscillations less powerful. They'll still happen but you'll have more time to get off the ground.