A few ideas for you:
You can use cast iron. Depending on what you are cooking this works quite well - but beware what you deglaze with, you might take your finish off or impart a taste to your food.
You can read the label on your rubber/plastic handle cookware. Many are oven-safe to 350F or so, the label should tell you.
You can use stainless steel or other all metal cookware. Paderno is my (Canadian!) favourite. Alternately go to your local restaurant supply store - so-called professional gear can actually be quite reasonable, because a kitchen has to buy so much of it.
Good Luck!
I don't know how to explain this well in words, but there's at least two different methods for flipping stuff in a pan, because you have either curved or straight sides on the pan. I personally find it easier to do in a curved sided pan:
I extend my arm forward, then quickly flip the back of the pan up while pulling back at the same time. The curved side basically acts as a ramp to propel the food up and over.
For a saute pan (straight sides), follow the technique that ElendilTheTall explained, where you tip it first.
A few things to remember when trying to flip your food : if the food's stuck to the bottom, it's not going to flip. Give the pan a quick shake first, to see if you need to stir to loosen any stuck bits.
And for the practicing -- besides just practicing with dry things first ... I'd suggest you start with just a little bit, get used to it, then add more ... and add more of something else, as with more than one color in the pan, you can see how well you're distributing things when you flip. (eg, lentils, other beans, rice, dry pasta, etc.)
Start with a mid-sized pan, as larger pans are actually more forgiving when catching, but too large makes it heavy and difficult to get a good flip. (ie, don't start with your cast iron).
If you have a grill, or a table outside to practice on, the cleanup's even easier. (although, use a cold grill, just for practice). You can also practice with just water, if you're outside, without as much worry.
As for when to do it -- lots of small things; stirring risks keeping the bottom bits on the bottom, and the top things up top, so things cook unevenly. Even if you use a spatula to flip, like you might for a larger item, the middle stuff stays in the middle.
The other advantage is that it only takes one hand -- I generally keep my right hand clean when cooking, and stirring sometimes takes two hands -- one to keep the pan still, while the other one stirs. It's also quicker to get a more thorough mix than stirring.
Best Answer
Skillets are not designed to hold much liquid; as you point out, the curved sides are optimized for easy flipping and turning (i.e. with a spatula). They also normally do not come with lids; I'm sure there are some out there, but even my All-Clad skillets didn't.
You can really only use a skillet for high-heat searing or frying due to their low profile. Don't bother trying to poach or even shallow-fry in one.
Sauté pans, on the other hand, have more height, and usually do come with lids. They're a little heavier, and don't make it as easy to turn, but they can also be used for shallow-frying and moist-heat cooking methods like braising or poaching.
I've seen people make sauce or chili in sauté pans as well; sometimes they're more convenient if you're only going to make a small quantity, since the large exposed surface area (relative to a saucepan) makes it easier to reduce sauces. It's a no-brainer if your recipe starts out with sautéing garlic, onions, vegetables, etc., and then has you adding the liquid ingredients; if you have a large enough sauté pan then you don't need to bother transferring to a saucepan afterward.
I think you've got it backwards; of the two options, a sauté pan is definitely the more versatile option. I use both, but if I only had the space or budget for one, I'd choose the sauté pan, because it can do everything a skillet can do (just not quite as well) and many more things a skillet is useless for. A skillet is really only necessary for people who do a lot of pan-searing.
Note that as commenter Owen suggests, there are "French skillets" sold which have higher bases; these are not really skillets, and are not normally sold with lids, but assuming you can find or improvise one, they would make semi-decent multi-taskers. I maintain that cast-iron skillets, while taller, are not nearly as versatile due to their reactive material and weight - I only ever use mine for searing/grilling.