There are two schools of thought on this:
- replace chain as many times as possible on a cassette until it skips
This is the more expensive approach but also results in fractionally higher efficiency from never running a stretched chain
- run chain and cassette together into the ground and don't care about chain stretch
Using this method the two can work well (i.e. no skipping) together for a very, very long time, and I've had over 30,000km using this method. But you ultimately do trade off a bit of efficiency and new chain "feel". Cheaper, but less mechanically efficient.
As with so many things in cycling, this subject has its fair share of lore, and many people think, wrongly, there is a huge downside to the second approach.
Guess which approach makes more money for the cycling trade?
Your secound choose being the more cheap may be valid if you use an Ultegra chain and cassette, but with the really expensive DA, SR cassettes, it seems to be more economical to change 4 or 5 40€ chains every 3000 miles than change the whole driventrain (250 to 350€) after 6000 or 7000 miles. I don't see a cassette last much longer with a chain stretching more and more...
Not in my experience - though I'll admit to almost always using Record and Super Record cassettes, and not spending all that much time in the Ti sprockets. Prior to that I used DA cassettes (9 speed) and got huge mileage there also.
It does work, but I would add I keep the drivetrain very clean - bike is quieter than most if not all other amateurs at about the same level of interest.
This subject is one where really YMMV due to different maintenance, riding conditions, shifting, etc. But the "keep riding the same chain and cassette" is too readily discounted as the wrong thing to do when for many people it would be cheaper, less hassle and perfectly adequate.
Stiff, Light, Aero - Pick Three!!