Yes, harvesting and reusing yeast is easy.
First, I separate yeasts into top and bottom croppers.
For top croppers you spoon the foam off the fermentation into a large jar, let it settle, pour into a smaller jar, let it finish fermenting, close the lid tight, and put it into the fridge (and if it's a diastatic yeast burp it regularly at least for the first few months of storage!). Top-cropped yeast lasts a really long time. I'm an outdoor brewer, and I just last week started using my top-cropped yeasts harvested in September/October.
Bottom-croppers you harvest, you guessed it, from the bottom. If you have a fermentor with a yeast dump valve, use that, but I don't, so I use a spoon after racking the beer off. You generally want the mid-layer of the yeast cake. The bottom is trub, and it feel significantly different. The top layer is late-flocculators, so unless you're conditioning your yeast for poor flocculation, avoid it. Bottom-cropped yeast doesn't last very long, probably because it sits for quite a while in the warm fermentor. I'd say four weeks is the max you want to store it (maybe more if you have a yeast dump valve), and two weeks is even better.
Bottom-croppers stay good for much longer if you use the "overbuilt starter" method, where you make a larger-than-necessary starter and save half. I generally don't use the technique, because it's extra work and cost, and instead try to plan my bottom cropping yeast brews so that they're reasonably back-to-back, or even directly back-to-back.
To use a harvested yeast, I decant the liquid -- for some top cropped beers it's close enough to the target beer that I drink it instead of wasting! -- and replace it with chilled pre-boil wort. Once the batch is ready to accept the yeast (boiled, chilled etc.), the yeast is usually already going and I pitch the whole thing.
You can pretty much always safely use yeast for a few generations. I like to use top-croppers indefinitely. At about 10 generations you need to start adding zinc to ensure full attenuation; I use servomyces which has a dose of 1g/hL, so a small packet lasts "a while" and is essentially free. Occasionally, if you start detecting sourness you have to wash the yeast with acid; it's not difficult, but a pH meter is a requirement. Of course, it's up to you if you find "yeast-hacking" worth your time and interest, and buying a new yeast after a few generations as "insurance" is a perfectly valid approach too.