The wonders of modern technology, eh?
Separating plutonium, uranium, and other products from spent fuel rods is a matter of basic chemistry - the different elements involve themselves in different chemical bonds and can be separated out that way, much like you'd precipitate iron out of a solution by adding the right chemicals.
The rods are left to cool for a number of months, then dissolved in acid under heavily shielded conditions. The various elements involved are then sorted out by chemical processes - I think one of them involves UF6, uranium-hexaflouride.
There is still vast amounts of potential energy in the "spent" fuel from a nuclear reactor, and the US policy of simply pitching it is like building a campfire with a big thick log, and disposing of the log when the outside bark is charred.
Uranium is 99.3% U238 and 0.7% U235. Pure Uranium is converted to UF6 where it can be enriched via gaseous diffusion (or ?) - and for reactor fuel - an enrichment of 3% to 5% is desired.
Assume a typical fuel bundle is about 1000 lbs - mostly Uranium - in an "Oxide" form - UO2 ... and zirconium filler and zirconium clad tubing. After about 3 years, the bundle is considered "depleted" and discharged permanently. (The reactor might have about 210 bundles ... so 70 bundles per year are discharged on a rotational basis.)
During operation in the reactor, assume about 50% of the Uranium is fissioned ... the rest is still useful, if you could reclaim it. But some of the U238 in the fuel absorbs a neutron and changes to Plutonium Pu239. Assume about 2% of the bundle is useful Plutonium when the bundle is discharged. 2% - 20 lbs. (10 kg). Now the actual details are more involved. The created Plutonium is also partially consumed during the reactor operations - and extends the "life" of the fuel bundle. And some of the Pu239 isn't burned but is converted to Pu240 (not good for bomb material) ... so "long-burn" fuel is less desirable for making weapons-grade Pu than fuel that spends less time in the reactor.
Both Japan and France recycle their fuel. They reclaim zirconium, reclaim Pu and U .... and the nasty fission products - less than 40 lbs per fuel bundle (4%) - is vitrified in glass for permanent isolation/burial. Even the U.S. was looking at this option . . . recycling the Pu back into reactor fuel ensures that there is no Pu "floating" around for a future theft for making bombs.
But breeders are designed to generate MORE fuel than is consumed .... and some concepts include mixing Thorium with the enriched Uranium fuel. You still get some Plutonium - but the Thorium "breeds" to U233 - which is also is hard to separate from U238 ....
So much wonderful technology is available ... unfortunately, the decisions on how to use it are often left to un-education idiots (like Jimmy Carter) - when they choose to make decisions based on politics rather than science.
Mike