Let's see if we can answer this in pictures. For one feed from the breaker panel, to have each of the outlets on their own GFCI, you want this:
![enter image description here](https://i.stack.imgur.com/L7K7W.gif)
For one feed from the breaker panel to have one GFCI protecting downstream outlets, you want this:
![enter image description here](https://i.stack.imgur.com/cRLPX.gif)
However, you say that you have a red and a black wire in the box, and this complicates things. You might have a MWBC (multi-wire branch circuit) feeding the outlets, with the red hot on one outlet and the black hot on the other:
![enter image description here](https://i.stack.imgur.com/qrC6L.jpg)
Or, you could have something else in this outlet, in which case (espcially if you don't feel confident) you really need to contact an electrician to get it sorted out.
The answer is: Don't.
Outlets are black, beige or white, and rarely dark brown or gray. Beige came into style by the 60s and white by the 90s. Yours are probably beige.
Leave the old receptacle boxes alone. They ARE grounded, there's nothing wrong with them, they obviously do have enough wire to reach the old receptacles. That's all you need.
"Modern" receptacles are the same size. Just change your receptacles from whatever's in there now, to a quality $3-4 each midgrade receptacle. If replacing backstabs, get receptacles with screw-and-clamp terminals - the wires poke in the back, but clamp by tightening down the screw terminals. Slick.
If the wires are ALL 12 AWG and the breaker is 20 amps, you can use 15/20A receptacles which have a "T" shape on the neutral slot.
If you had "back stab" types before, don't cut them off because you'll lose precious wire length. Instead, pull steadily on the wire while twisting the receptacle back and forth (not enough to deform the wire), and the wire will ease out. It will have scratches but will work fine with screw-and-clamp or side terminal screws. Don't use it again on a "back stab", in fact never use backstabs ever because they stink. Also don't pry it out with a wire nipper because you'll nick the wire.
GFCI
Ground faults are current following abnormal paths, such as through a human who is being electrocuted. Mainly, they protect people and pets from shock. Usual cause: bathroom mistake or defective plug-in appliance.
If you want GFCI, remember the first rule: Only the first GFCI device needs to be GFCI. Everything downstream can be plain old outlets, and they inherit GFCI protection because they are fed off the "load" terminals of the first GFCI. That's how the GFCI can be a circuit breaker. It can also be a new receptacle you insert at the front of the chain, for instance 1 foot from the service panel. I do that all the time. GFCI receptacles are cheaper than breakers.
Honestly it depends what threats you're worried about, but if you're distrust your old wiring, or want to follow the latest code, I'd consider AFCI instead, or both.
AFCI
Arc faults are loose connections, broken wire etc. causing current to leap across a gap. This makes a lot of heat and starts fires. If you've heard the unique "sizzle", it also sounds like that on the wires. AFCI's have a computer inside which listens for that peculiar electrical noise. Usual cause: defective in-wall wiring or receptacles; sometimes bad appliance.
Generally the cheapest way to get AF protection is with an AFCI circuit breaker.
Combo AFCI+GFCI
You can also get combo AFCI/GFCI breakers, but they are expensive. Watch for ones which will indicate why they tripped, otherwise you'll go bonkers.
Another way to get combo AFCI is to use an AFCI breaker, but a GFCI receptacle in the first point in the chain. This makes it more obvious what tripped.
Best Answer
Do not use lamp cord to extend a circuit to a new outlet! The wire is neither big enough, insulated enough, or grounded. It's illegal everywhere, and I wouldn't consider it to be safe.
Use Wiremold with standard electrical wire. That's why the product exists.