Finding a riding position that works for you is mostly an exercise of trial and error. It's mainly a matter of preference and there are no hard and fast rules; however, there are riding positions that work better for particular situations. Road bikes typically have their handlebars below the level of the seat; this hunched-over position is better for maximum power off the line and provides lower drag at high speeds. Mountain bikes with their short flat bars slip through tight places and keep your weight forward allowing the wheels to steer over rough terrain. Touring and trekking bikes need to have handlebars that allow for multiple positions for climbing, descending and spending long hours in a headwind. And what about those impossibly short fixie straight bars, you ask? They're good for "smashin' through traffic", apparently.
City bikes typically have a more upright ride with more weight on the saddle and less on the handlebars. This combination makes your back straight and more upright giving you a good view of the road ahead. Because you're often riding more slowly you're less concerned about aerodynamics than you are about seeing over cars or being seen. Bikes with this type of geometry are more forgiving for beginners or those carrying varying loads. For most commuters who are dragging along laptop cases, groceries, and maybe some morning coffee an upright ride makes sense. Call it the Riding Position for the Rest of Us, if you like.
Where can I get an upright ride?
If you're buying a new bike, make sure it fits. Yes, it seems obvious, but you'd be surprised how many people settle with a bike that doesn't quite fit because what the LBS had in stock or on sale. Many bike manufacturers only offer one or two sizes, so keep that in mind if you're not perfectly average in height. A properly fit bike, regardless of the type of handlebars, angle of the seat tube, etc, will feel better. Most of the bikes that we feature in Bikes for the Rest of Us have a fairly upright riding position (some more than others). Go the manufacturer sites, look up the dealers and go find a bike to test ride.
A common scenario is retrofitting a bike that you already have (or got from your Uncle, local garage sale, etc). There are some techniques that can help you make the bike more upright and appropriate for city use. Most of these solutions are fairly inexpensive and are in order of effort (but if you're going to replace your handlebars anyhow start there). Keep in mind you may have to extend your cables, particularly your front brake, if you raise your stem by more than an inch. Note: My stem examples below are for quill stems.
Raise the Handlbars
If you like your handlebars, but feel you need to get them a little higher, try raising your stem. Sheldon Brown tells you all about how to do it here.
Replacing the Stem
When you raise your stem (quill stem) you may see a line that says "minimum insertion depth" or something similar. If you see this mark the stem can't be extended any more and you may have to get a new stem. Prices and quality vary, but a good bet would be something like the Nitto Tall Stem or a Wald 511.
Get an Extender
If you need to raise your bars more than a couple of inches then consider a stem extender or stem riser. This device allows you to keep both your existing stem and handlebars, but the catch is that it's going to raise everything at least two inches. The appearance may also be a concern, but the price is right.
Replace the Handlebars
If you're not happy with your current handlebars you can get new bars that have some rise to them. Most North Roads or All-Rounder type bars have a 1-2" rise. You can find handlebars with a similar shape that have up to an 8+ inch rise. Wider bars give greater control to laden front baskets. Narrower bars like the VO/Nitto Montmarte allow you to squeeze in between tighter spots. The trick with handlebars is trying out a few, which takes time and, of course, several handlebars. Be sure to do your homework if you plan on reusing your grips, shifter or brake levers as there are different diameters of handlebars (see section on compatibility issues here).
Most of the links that I showed you are for online retailers. You may want to start with your local LBS, Bicycle Club or garage sale to see what you can find.
Photo Credit: Sketch above by Linn