Different kinds of crystals are cut in different shapes, so they resonate in different ways and have specific temperature, stability, or aging properties. The 'AT' cut is the most common, and the best advice for a beginner is "use what it says on the circuit diagram unless you know specifically why you need something else."
A crystal sitting on two capacitors is called a 'Pierce oscillator', and is the most common kind of crystal oscillator. The size of the capacitors has some influence on the frequency (stability, temperature response, etc), and some people will play with the values to get specific results. Again, the best advice for someone new is, "use the values from the circuit diagram until you know how to build something else."
Ceramic resonators are all-in-one devices. You plug 'em in, power 'em up, and watch 'em hum.
The big difference between crystal oscillators and ceramic resonators is frequency stability. Basically, crystals are really good.. a decent crystal oscillator's frequency will only drift from its average value by a few parts per million. A ceramic resonator will drift by about .5%. An RC oscillator (like the chip's built-in clock) usually drifts by about 5%.
The big question is "how much accuracy do you need?" Crystals are good for communications gear and timekeeping, because the whole point of the game there is to be stable, predictable, and reliable. For a microcontroller that isn't doing hard-real-time work, a ceramic resonator (or the chip's internal oscillator) should be just fine.
If you're working on a breadboard, you don't need much accuracy.. or perhaps more correctly, you won't get much accuracy.
Breadboards are notorious for having lots of parasitic capacitance, and for causing all sorts of problems at high speeds. Analog design guru Bob Pease used to call them "white blocks of pain". Putting a crystal oscillator in a breadboard would be like trying to use an electron microscope in the back of a pickup driving down a gravel road.
Honestly, you'd be just as well off to forget about an external oscillator entirely and use the one built into the chip, at least as long as you're working in the breadboard. If you find yourself needing more frequency stability than the chip can provide for itself, you'll have to switch to another construction technique to get it.
When you void a product warrany, 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.