The age and features of the handset(s) involved will dictate what you can do here, but it's good that you're doing this, as phones should be treated like PC's in the way we say about backing them up, for the very reasons you have said, there are several ways this can be done, but the phones connectivity is the crucial point.
If you list the makes\models of what you want to extract the data from, I will be able to be more specific, but the ways to do this are:-
Phone Swap; a utility on new phones that walk you through importing your data from the old phone, but the old phone has to be compatible for this.
Software; many phones have software that they work with for syncing the phone and the like, but there may be phones that do not have this.
Online\Cloud; some phones can backup their contacts etc. online, but this is probably one of the options that very few makes offer, but I know Nokia phones do via Ovi.
Physical Connection\3rd party Software; basically being able to connect to a PC via a lead or similar, and using 3rd party software to get the data.
Exporting; simply export the phonebook in a csv file.
Saving; simply save the phone book to the sim card, then put that sim card into the new phone and import them from the sim card, but some of your phones don't use sim cards, and a lot of phones are now using micro sim cards, so the old sim card won't fit the new phone.
Memory Card, you can save your contacts to the memory card if it has one, which can then be placed into the new phone, but again is phone dependent.
Google; as per Online\Cloud above really, if you can access your google account on your phone, you can sync with them, probably the best option for many, as it does not depend on a specific phone, you just need to be able to connect to it on your phone, the good thing here is you won't have problems where phone makers tie you down and make it hard to transfer to a different make of phone.
The downside is privacy, google want to connect the whole world together, and by using them they then take your phone book, and start joining the dot's, introducing others you may know, and will allow you to search for others (old school friends etc.) if you allow it, personally I don't have a problem, but I do find it annoying sometimes the way it try's to push you for more details to connect even more dot's, but if privacy is a concern for you (many do, often out of principle more so), then this way is probably not for you.
I'm sure I have missed another option, but can't think of it for the life of me, anyway let me know what you have got, noting which phone is to be saved\exported and imported, and where you would like to save a backup, then I can hopefully be more specific for you.