Assuming your interest is first and foremost in pictures, and not some other aspect of the photographic hobby, then the best way to make pictures that satisfy you—that "nourish your enthusiasm," in the quaint and kindly words of Uncle Ansel—is to spend time with a camera in your hands, preferably in an opportunity-rich environment of the kind you have a feel for (whether that's the landscape, city streets, a foreign country, or your living room). The second best way is to work on your technique—your shooting technique primarily, but your processing technique too. Only the third best way is by acquiring a camera or lens that does something your old ones can't.
In terms of cameras, the single biggest jump you can make, in most cases and for most people, is to move from a small-sensor (chiefly, 1/1.7", 1/1.8". and 1/2.5"—the last being the smallest) camera to a large-sensor (4/3 to "full-frame" 24x36mm) digital single-lens reflex (DSLR). The DSLR is currently the form of most higher-level, higher quality cameras, and the lens choices are customizable to your needs and have much more variety. The primary difference between point-and-shoots and DSLRs, however, is the size of the sensor: digicams have sensors roughly the size of a fingernail, and DSLRs have sensors roughly the size of a postage stamp. It’s a huge difference, and it shows in overall image quality at higher ISOs (the setting for light sensitivity) in lower light, and in overall smoothness, color rendition, and in the amount of processing you can apply before things start to go awry. Unless you always shoot in good light, our top recommendation for anyone currently shooting with a small-sensor digicam, assuming they can handle the size, is to make the jump to large image sensors. The cheapest and most satisfying way to do this is to invest in an entry-level DSLR.
Our top pick at the moment is...
...the 10-MP Olympus E-420. Yes, I know that Olympus has just announced a successor, the 10-MP E-450 (although the difference appears to primarily be one digit in the numerical appellation—I do not have the true gearhead's patience for minute differences among nearly identical models). So, why the E-420? Simple. Look at this price. That's with a copy of Olympus's (very good) standard lens, which honestly would not be the worst deal in the world for that price in and of itself, if we hadn't gotten spoiled with loss-leader prices on kit lenses already. That's a class-leading bargain and, as I probably don't have to point out if you're not thirteen and you still remember what DSLRs cost ten years ago, an amazing price in general for such an appealing, portable, and capable camera. That deal won't last long, so the shelf life of this recommendation is limited.
The E-420 is also extremely small and light (almost as much so as yesterday's camera, really) and it has a compact prime normal lens available for it that fits perfectly with its small, light aesthetic. (This lens with a Nikon D60 is also a fine combo.)
But really, there's no particular reason for choosing one entry-level DSLR over another. They're all good—very good, actually, since they have economies of scale and intense competition for a large market standing behind them—and none of them are perfect. The market leaders are the Nikon D40, a good 6-MP camera which I don't recommend at the moment because of the lens(es) it's currently bundled with, and the 10-MP Nikon D60, and the 12-MP Canon XSi, recently replaced by the announced-but-not-shipped 15-MP T1i (that's tee-one-eye—Canon has trouble with product names). Both the linked cameras are recommended with the lenses at the links; without those lenses, the cameras slip a bit in the recommendations, despite their market-leader status, because they won't have image stabilization (IS) (Nikon calls it vibration reduction (VR)—it's the same thing), and are thus somewhat crippled in comparison to the competition.
In fairness, the value-leader E-420 doesn't have stabilization either. To get that in the Olympus line, you need to go to the E-520, which I recommend you not do. Why? Because the replacement 12-MP E-620 has been announced, and it looks like more than a name change. Olympus has been on something of a roll lately, letting the research and development of the top-of-the-line E-3 spill down into successive tiers of less expensive models. The E-620 will be worth waiting for in preference to choosing an E-520 now.
The entry level DSLRs with body-integral image stabilization have an advantage in that they provide IS with any lens. At higher price-points we can argue till the cows come home about body-based vs. lens-based stabilization, but at the entry-level it's kind of a no-brainer—get it however you can, and the best way, unless you want to stick to the inexpensive kit lenses Canon and Nikon provide it with, is to get a camera that has it in the body. These include, currently, the 10-MP Pentax K2000, the aforementioned Olympus E-620, and the 10-MP Sony A200. As always with an interchangeable-lens camera, look at the lenses first, and make sure the brand you're looking at already has the lenses available that you want and can afford.
So why start out with a bare-bones entry-level model, when each of these manufacturers makes slightly higher, slightly better-specified models for not much more money? Well, that's a philosophical question. Being basically teacherish in my outlook and motivation, I encourage people to use their purchases to learn about themselves. If you are currently using a small-sensor point-and-shoot, then the biggest single jump you can make is to a larger-sensor DSLR. I do think you should do this, but I don't necessarily think you should buy a more expensive camera first. Get an entry-level model and plan on keeping it from one to three years. Explore it; take lots of pictures; either you'll learn that you love the small, light, consumer-level camera style, in which case you should replace your camera with one of the higher-spec'd lower level models, or you'll be frustrated often enough by the inherent limitations of the basic models and using your camera so much that you'll know you want to move up one, or even two tiers and get a more deluxe model. It's not just a shopping game; it's a learning game too.
Monday: #5, as our countdown of the "T.O.P. Ten Recommended Cameras" continues.