Just bought old house, only some plugs are grounded
Good you're asking now.  So much has changed in the last few years. (Really, no joke).

Why do we want grounds?

Grounding was a frantic attempt to solve several problems from human-made electricity.  At the time the only protective devices were overcurrent protection, in the form of fuses or breakers.  

Wired "ground" is pegged to neutral and earth in one place. Neutral is the return for human-made electricity.  Earth is the return for natural electricity. 


Arcing across loose connections or damaged wire causing house fires.  The hope was it would arc to ground instead, giving a very high-current path which trips the overcurrent device.  
People being shocked by wayward current trying to return to source via water pipes or the earth.  Providing a very high-conductance path back to source should divert almost all fault current, and again hopefully trip the overcurrent device. 


It helped with natural electricity too, which wanted to get back to its source which is earth.  External tower antennas were already grounded, but some lightning energy would make it down the signal wire to the radio or TV. It was discovered grounding those would help stop fires by giving lightning a good path back to its natural source.  The same applied when they found new electronics were vulnerable to shuffle-feet-on-carpet static electricity, now called ESD. 

What changed

For shock hazards, GFCI (Ground Fault Circuit Interruptor) aka RCD (Residual Current Device) became established over the last 30 years. This trips on differential currents.  In theory, all the current going down the hot should return on the neutral - currents should be equal.  If not, current is traveling a third path in what's called a ground fault. That could be through a human.  A 

GFCI can only detect a fault in its zone of protection, which is all the premises wiring downstream of the GFCI.  Europe protects the whole house with one RCD, but uses a weak 30ma threshold - which will still allow you to be stunnned and then fall or drown. That's why America uses the 8ma threshold, one circuit at a time because 8ma is too sensitive for a whole house. 

For fires from arcing faults, AFCI (Arc Fault Circuit Interruptor) have become affordable this decade.  They contain a small control computer which does acoustical analysis of the power line, listening for the telltale sounds of *arcing - if you've ever hooked up speakers while the stereo is on, you know the sounds.  Its zone of protection is all downstream wires and a little bit upstream (it can hear arcing nearby). 

Electrical fires are the #5 cause of house fires and the #3 cause of fatal house fires.

GFCI and AFCI breakers (and even GFCI+AFCI Combos) can be had for any modern panel - not Pushmatics (handle those by adding a modern subpanel), FPE or Zinsco (both of which are dangerous and should be immediately replaced!) 

Third and also important, in the 2014 NEC (effective 2015-16), they greatly liberalized the rules for retrofitting grounds.  Now you can run just a ground wire to almost anywhere that needs a ground.  Ground paths can be combined for all circuits out of a panel, as long as each point gets a thick enough ground path. Install the 10AWG backbone first to major appliances, then all the 12AWG grounds to the nearest 10AWG place, then 14s to the nearest 10 or 12.  What's new is they made it easy. 

You cannot use building steel or water pipes or the grounding electrode system  for the grounding path, the destination is the ground bar in the panel. 

 Problem                    Weak solution          Strong solution

 Human shock             Ground or 30ma GFCI          8ma GFCI
 Fires from arcing       Ground or 30ma GFCI            AFCI
 Lightning                        -                    Ground
 ESD                              -                    Ground


To protect your family

You can now see that burning all your money pulling new cables is the wrong thing to do, a waste of funds needed elsewhere. 

AFCI+GFCI are the place to start.  

It would be a waste of money to spend $40 on an AFCI breaker and then $20 on each of its six receptacles for GFCI+receptacle combo devices.  As discussed, GFCI's and AFCI's cast a "zone of protection" to all devices downstream on that circuit, including the wiring.  A $50 combo/dual-mode GFCI+AFCI breaker will protect all outlets on that circuit, including hardwired outlets. The outlets need to be marked "GFCI protected" and if applicable "No equipment ground". If you don't like the ugly stickers that come with GFCI devices, you may use your own marking methods, I like white P-touch labels on white covers. 

GFCI "receptacle" combo devices are almost always installed wastefully, even by pros - wasting money and forfeiting protection that is available.  Don't use them unless you really bone up on how the "downstream protection" works, that is the essence of the devices.

Now if you have equipment that you want to protect from static electricity, you can retrofit grounds to that location.  Retrofitting grounds and fitting AFCI and/or GFCI are orthogonal and can be done independently without impacting each other.

The dryer and range

These should be on your "ground it" list because Code specifically allows a very ugly method of connecting dryers and ranges without a ground using the toxic NEMA 10 plugs: bootlegging ground from the neutral wire.  That's all grand until the neutral wire develops a flaw: when you turn on the dryer or open the oven door, it electrifies the chassis at 120V.  The correct fix is the modern NEMA 14 type plugs, and jumpering the dryer/range to separate neutral and ground.    A GFCI breaker will be an effective substitute, but expensive.