Well, nowadays the manicure kits can be a lot more complicated than that. Mine for example is probably the most elaborate of its types. I use a small silver train case as my manicure kit. I have all of the basics mentioned above, except a cuticle pusher. I never much enjoyed that part of an at home manicure anyway. I did replace the cuticle pusher with an orange stick which accomplishes the same basic thing. I also have some cotton, fingernail polish remover (I have both the bottle and the jar that you stick your fingers into), sticky tapes for fool proof at home French manicures, tons of different polishes, and of course one of those cute little toe separator thingies. Mine is hot pink and has palm trees on the end. This is a manicure kit that would make my Grandma proud.
In addition to the old fashioned manicure sets you can purchase all kinds of kits at all different kinds of stores. Some kits will have cute little carrying cases like mine (well not as cute as mine, but cute nonetheless). There are kits aimed at the target market of teenagers and small children. You can even purchase a manicure kit with Dora the Explorer on it. There are also more sophisticated kits that cater to an older more sophisticated target market. These can be compact for the frequent traveler or huge and strictly home bound like mine.
If in doubt, you can purchase one of these preassembled kits and then gradually add to it. You can get all of the basics in the beginning, and add new pieces as you need them. After all, who doesn't need one of those cute little toe separator thingies.
It is a good idea to go through your manicure set periodically and purge any old and outdated manicure supplies. You should replace the orange sticks and emery boards pretty regularly in order to avoid spreading fungus and bacteria that live under our nails. In addition, you should throw out polishes after about a year. I am guilty of adding fingernail polish remover to that last favorite bottle of discontinued perfect red. You know this particular beauty sin; the polish is real thick so you thin it out using remover. Well stop doing it, its ridiculous. Buy a new color already!