I do this an awful lot, I drink a lot of tea, and since I started getting better quality loose-leaf instead of teabags - I wanted to get the most out of my tea. My daily mug is a three-cup one, so I'm used to brewing that amount as well. I can usually get three mugs brewed tea, and two more boiled - though some brands will give three to five steeps boiling, for 8-10 steeps for one basket of leaves.
For the first brew, I follow the directions exactly - if you're using an amount measured for your water, there's no reason to change. I have a few times used more, with the intention of steeping less time and more often - not really a good idea, it tuned quite bitter and overly strong even when steeped for a very short amount of time (seconds, for the second steep). I have also used less, which sometimes works alright (especially for very strong teas) - but I do tend serve mine sweet and also to be fairly tolerant of variations in tea, from milk tea to barely colored water, so your preferences may vary. But one teaspoon per cup, and the recommended brewing temps, should be a good amount to start with - as long as you recall you can monkey with either to suit your own preferences.
On each subsequent brew, you will want to steep the leaves somewhat longer. Since I usually am brewing at my elbow, start with the time I took for the first brew - and start checking for color, then for scent, until it's reached a desirable level of flavor. This can take varying amounts of time depending on your tea, so I don't time it (not like, two extra minutes or something - though it you brew consistently you can probably figure out your preferred times), but I always judge by color and taste. It may not taste the same, you understand, as your first cup - but it can be an acceptable cup of tea on its own.
I think cultural customs that rely on multiple brewings tend to steep very lightly and shortly, and use a lot of tea relevant to the water - and the tea certainly changes in flavor over several brewings, as the different compounds dissolve out of the tea, which to them is a positive attribute. For those that like a full flavored cup, or a consistent one, there will be fewer but stronger brewings of tea, and less tea per water at temperatures and times designed to get the most out of the tea so it is more easily controlled (this is where your "expert instructions" come from, calculated to brew the most out once not maximize rebrewing) - different culture, different expectations. I also don't think black teas hold up very well to this kind of overloading and understeeping, they dissolve flavor quickly and tend to become bitter for me - but teas with more delicate flavor to being with, greens or puer-eh or even some herbals, will probably hold up better.
For myself, once I've reached the point, usually on the third or fourth steep, where the tea hasn't reached the point I'm pulling the leaves before it cools to drinking temperature (three cups of boiling water in a thick mug cools slowly, okay), I set the tea leaves aside, and brew once or twice on the stove top - an extended boiling water infusion, five or ten minutes each. It doesn't get bitter, not the way fresh leaves do, since the prolonged steeping has already drawn out much of the flavor compounds, it tastes - like a teabag, kinda weak and generic, usually. Honestly, the last steep is sometimes little more than colored water (depending on the tea). But others have noted bad experiences with boiling infusion tea (depending on expectations), so again your preferences may vary.
As for teas that are good for multiple infusions - green teas are often touted as very good, especially ones like gunpowder where the tea is tightly folded and packed - so it's still unfurling during subsequent steeps. Puer-eh teas are excellent, especially in brick form - the tea takes time to hydrate, but it also brews strongly and so lasts several steepings easily (and it tends to steep sweet, that is, not get bitter easily unless you get the raw stuff). Maybe watch your brewing to taste, the directions are sometimes quicker and cooler than makes a comfortable tea for me (45 seconds of brewing does not make a strong tea to my taste unless vast amounts of tea are used, like a third of a cup). There's some blend of black tea that brews like ten times per serving - I don't actually know what it is because I found it in a set of sample blends, so they're advertising the blend mix, not the individual tea leaf. Herbal teas - yes, steeping like tea will give several good infusions since the flavor extracts slowly - some will even make several boiling infusions (and again, they don't get bitter even if steeped long or hot).
Keep in mind that what you consider a good cup of tea might not match other's expectations. I usually find Japanese and Chinese style teas 'delicate' and not flavorful enough (brewed at precise low temps and quick times), and brew at higher temps and for longer - while obviously their method is a lot easier to get multiple brewings out of, since they're getting less flavor extracted per serving, and I'm burning through two or three of their 'brewings' per brew of my style. On the other hand, south Asian teas (India, Thai, Middle East) tend to make strong bitter infusions by boiling the tea, and mellowing the flavor with sugar and milk and optionally, spices. These will not rebrew at all, since the extended boil is intended to extract everything, everything that could be in your tea leaf and waste nothing.
My five infusions per serving is based on a medium weakish tea, flavorful but not boiled strong - and it definitely progresses in flavor over the series of brewing, becoming... more generic in taste? like I said, the first is a good quality loose leaf, the last is a generic teabag... I end up with five instead of three because I brew a bit lightly and I will still drink the later, weaker brews. Someone who brews more strongly or doesn't like the weaker flavors will get three brews. Someone who does both - might struggle to get two brews from the leaves.
Are you wasting tea by your method... again, it depends on your brew method. If you're brewing to instructions, then no it isn't wasted...it is just spent. If you're doing a boiling infusion style tea (where you want strong, bitter flavors), you can definitely use less tea and higher temps (boil it), or longer times (oversteep it) for very nearly the same effect. Once your tea leaves have reached a point where you aren't getting a good cup out of them, you aren't wasting them by discarding. There might still be a bit of 'flavor' in your leaves, but it might be more trouble than it's worth to you to extract.
If you're really determined to get the last of the tea flavor out of your leaves, and your sequential brews are too weak for your tastes, you can try adding more tea. A teaspoon, or even a half, of fresh tea added to your second brew of leaves might make a cup that is comparable to your first cup - and at four teaspoons for six cups of water, you're still coming out ahead. Or, if you drink the same (or even similar) teas a lot, you might be able to brew your tea leaves in a larger batch on each go - maybe two batches of twice-steeped tea will make a single batch of thrice-steeped tea (6 tsp per 15 cups if you get two brews before combining).