IMHO the best, fastest and most effective, way to learn the basic knots is to attend a class.
I have taught a short course as part of a spelunking trip and found that two hours of rope work, and fun, and a quick ten minute refreshed before going in was effective. Even years later quite a few of the people remember most of the knots. Learning in person makes a lot of things clear that might get lost online or on paper.
Perhaps the biggest thing to learn is how to 'dress' a knot. A marginal knot well dressed is often better than a great knot left undressed.
Working from memory I included:
Square knot - include warning, demonstration of failure by inversion and using half-hitches for safety.
Bowline - include use of half hitch as safety. Teach one-handed tying if time allows.
Sheet bend - also double sheet bend.
Clove hitch -include warning of it rolling loose and half-hitch for safety.
Round turn and two half hitches
Rolling hitch.
Swiss seat.
Bowline on a coil.
Alpine Butterfly - teach both 'twist' and 'coil' methods of tying.
figure eight stopper.
There is always controversy over which knots to include, and which to leave out. Everyone has their favorites and knots they dislike. I like this list because it is short, the knots are simple to tie, most can be ties by touch alone, easy to understand, and anyone who masters the list can do most things effectively.
Of course there is some redundancy in even this short list. Experienced knot people will immediately spot that the sheet bend and bowline are the same knot. Or that a rolling hitch is a clove hitch with an extra wrap.
The bowline on a coil is a bit esoteric but was included because we weren't using harnesses. The drops were small and the rope was used as a safety while they climb up and down.
The alpine butterfly is strong, handy and fun to tie and fun to teach but a slipknot and half hitch would work as well in most fixed rope cases. I include the butterfly because every time you tie it you feel like you did something.
The rolling hitch doesn't have much use in climbing, prussic would be better, but it is just so useful in every day life that I put it in. It's important if you sail.
Trucker's and timber hitches are good additions. The Flemish bend is a good one.
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