Technically it's possible, but the color change you'd get would look rotten.
RGB LEDs are in fact three LEDs in the same plastic container. It takes a certain amount of voltage to make an LED work (called the 'forward voltage'), and the color of the light coming out of an LED is related to the forward voltage. Red LEDs have forward voltages around 1.7v to 2v, green LED have forward voltages around 2.5v to 2.7v, and blue LEDs have forward voltages around 3.2v to 3.4v.
Hooking all of those to the same solar panel would give you a weak red glow for low light levels, a moderately bright yellow for medium light levels, and yellowish-white for high light levels. The changeover points between colors would be fairly abrupt, and in general the output wouldn't be what you'd hope to see from an RGB LED.
You'll get more predictable results from a single-color LED and a resistor, but even that will give you mediocre output. The trouble isn't with the LEDs, it's a side-effect of the way the human eye handles light. The amount of light an LED puts out is proportional to the current flowing through it. Twice as much current gives you twice as much light. The trouble is that "twice as much light" doesn't mean "twice as bright" to the human eye. In fact, the change is barely visible.
That happens because the human eye evolved to see both at night and during the day. Our retinas are sensitive enough to register a single photon in the dark, but we also have to deal with full sunlight during the day, which is several million times more intense. If our vision was linear, we'd either be blind in full sunlight or blind in moderate shade.
Translating that back to solar cells and LEDs, you'll be able to make the LEDs dim by reducing the amount of light that falls on the solar cell, but the dim-LED-to-bright-LED range will occur over a relatively narrow range of light levels on the solar cell.
To make the experiment more interesting, get a selection of single-color red, green, and blue LEDs, and a selection of resistors with values of 10 ohms, 100 ohms, 1000 ohms, 10,000 ohms, and 100,000 ohms. Make your initial connections to the solar cell in a way that lets the kids compare different resistor/LED combinations and light levels. That will make the differences in forward voltages and current levels easier to see.
As for soldering: you'll want an iron, a wet sponge to clean the tip, some flux, and some 60/40 lead-tin solder. Lead-free solder is significantly harder to work with, and the risks associated with 60/40 are negligible as long as you don't wad it up and chew on it. Contrary to what you may have seen or heard online, soldering fumes contain no lead whatsoever.
Pure lead doesn't even melt at soldering temperatures. Solder is what's called an 'eutectic' alloy, which means it melts at a lower temperature than either of its components.. rather like salt and water. The lead and tin don't melt so much as dissolve into each other. The vaporization temperature of either metal is far
above soldering temperatures, so the risk of exposure to lead vapor is nonexistent. If you're concerned about skin exposure, you can make 'solder pencils' that prevent direct contact: http://info.yawp.com/stuff/solder-pencil/s1-make.html
The smoke you see during soldering comes from organic molecules burning off, most of which come from the flux that cleans the metal while it's hot. Electrical flux is made of rosin, which is the solid component of pine sap. Soldering fumes are basically refined wood smoke.. not the kind of thing you want to breathe in large quantities on a daily basis, but the amount you'll get from an hour of soldering is less than you'll get from five minutes of lighting a campfire.
When you void a product warranty, you give up your right to sue the manufacturer if something goes wrong and accept full responsibility for whatever happens next. And then you truly own the product.