I sit as upright as possible. You don't actually want to sit totally upright, because then every bump in the road is transmitted straight up your spine.
For a maximum upright position and down to a semi-sporting position, the best bars with the most ergonomic grips is North Road bars, which bring the grips around towards you, up, and then angle them downwards so that you're riding with straight wrists. (Well-specified and -fitted drop bars do the same when you're riding on the hoods.) As John R has explained above, you adjust reach length by stem length. But it isn't always necessary, because as your upper body moves more vertical and your hands thus move backwards, the North Road Bars also reach backward to meet where your hands fall in the new body posture.
Start by putting your saddle in exactly the optimum position in relation to the pedals because sitting upright will lose some power relative to drops, or require more power because of adverse aerodynamics compared to a flat back. Then take the position in which your back will be comfortable over the long term and do not compromise it by one millimetre -- there is a combination of stem and North Road Bar that will match your preferred posture to the steering tube.
Nor is your position now fixed forever. If you choose the North Road Bars right first time, and don't make any extreme installation decisions when you're doing it for the first time, the whole of your body on the bike and the bars and stem operate like a pantograph, so that within reason there is a range of positions possible without further investment.
It's less difficult to get a good fit than it seems. I did it three times in a row by mail to countries two sea passages and several borders from where I live -- after several times failing to do it with local LBSs...
Ask in the first instance to see a set of Uno Kalloy North Road Bars. They come fitted to a lot of Taiwan-made bikes, and are then taken off by LBSs to "add value" by fitting more expensive bars, so many LBSs have a pile out the back. They're also a good value if you have to buy new. They work with the Uno Kalloy angle adjustable stem with a lever for toolless changes on the road -- an added comfort function I discovered on my Gazelle Toulouse which came with Gazelle's own proprietary toolless adjustable stem. Uno Kalloy North Road Bars are desirable because they're good, they have long grips that you're supposed to saw to size yourself, and if they haven't been sawn short they're ideal for Rohloff rotary controls right under your thumb without lifting your hand from the grip. They're available in aluminium and steel; I like the steel ones because it is a pain to replace the ali ones every five years or whatever.
The more upright you sit, the wider your saddle can/should be, though "wide" is a personal choice. I like the Brooks B73 for a fully upright posture; some others here see no need to go bigger than the Brooks B67.
Good luck with achieving a painless posture on your bike.
