Sample tastes from day 1. There is nothing about the fermentation process that will make you sick or taste disgusting and tasting your wines and ciders throughout the process will help you better understand from the "inside" what is happening to your fruit.
Others may disagree but the number of stages in the process is really not something that is objectively definable. The stages have as much to do with the wine maker (or cider maker) as it does with the fermentation process itself. Assuming you are ignoring the preparation of the juice (pressing, for example) or prep before you pitch the yeast (enzymes and K-meta) then you will ferment vigorously in a primary which does not need to be sealed from the air - indeed, quite the opposite as yeast requires O2.
If you bottle immediately then you bottle immediately but many cider and wine makers prefer to age their fermented liquors in air sealed carboys. This is the same process as the first mistakenly referred to as a secondary fermentation when the only thing that is secondary is the fermenter (the vessel) and not the process itself.*** It would be into the secondary that you would add any spices or additional fruit or even additional acids and tannins (they will be blown off in the primary).
Using a secondary has the advantage of allowing the wine or cider or mead to age off the lees or sediment which in some circumstances can cause off odors. It also has the advantage of assisting the wine to give up its vast collection of CO2 molecules. And so..
many wine makers and cider and mead makers don't simply rack once but two or three or four times in the course of 6 -12 months, each racking removing more and more of the sediment and fruit particulates that create cloudiness and so they have a brightly cleared product, very still and very dry with virtually no yeast in the wine or cider.. Is that two stages or three? or half dozen? I don't know. You decide. But then there is bottling and some wine makers and cider makers rack to a bottling bucket and bottle from there. And then there are those who transform their still dry wines and ciders into sparkling wines (adding more yeast and sugar and riddling)... Bottom line - and again I am sure that others on this forum will disagree strongly, but for me there are three stages - fermentation, aging and bottling - and as the great French film maker - Jean Luc Godard said -"Every story has a beginning, a middle and an end... but not necessarily in that order" , and so it is with wine - fermentation, aging and bottling although not necessarily in that order.
*** To confuse you even more (and I hope I am not doing that, GoingInCIDER), there is actually a secondary fermentation but that does not involve yeast. This is when the malic acids in wines and ciders are broken down by bacteria to produce lactic acids. Lactic acids are far less strong than malic and so the cider or wine tastes far smoother. Often, the way one ferments will prevent this malo-lactic fermentation from occurring and sometimes the chemicals that wine makers add to prevent additional fermentation from occurring when they add more sugar to sweeten the wine will create geranium off flavors in the wine or cider if the MLF (as it is often called) takes place spontaneously. But if you allow cider or apple wine to age 9 months in a carboy or a bottle the flavor changes significantly and the cider or wine seems to be transformed...