Dear Nicojo,
probably, the best and easiest thing is to keep everything in your mind and have someone else's as backup :-).
That means, in practice (at least in my experience) record keeping does not work as well as it should unless there is a very insisting lab manager who is keeping an eye on everything and everyone and is able to keep track of it for a longer time.
In a lab where I once worked as postdoc, this was very well organized: The chief technician kept records where the master stocks of cells, plasmids and primers were. If you were to use something new, you would obtain an aliquot and propagate it by yourself for your needs, if necessary. Orders of new primers were only accepted when you presented a database sheet where you had entered all details (concept, sequence, application, PCR parameters etc.). When a student / scientist / technician changed the lab or left the institute, (s)he had to provide documentation on all new plasmids and cell lines (s)he made. Of course, all this paperwork was a bit tedious, but it was easy to retrieve all the necessary information when you started to work with something new and -surprise- the information was quite reliable.
Basically, as PI you need to set up appropriate rules, everybody simply has to follow. Don't give anyone a degree or a testimonial unless (s)he has completed all the paperwork. I know this sounds mean, but in my experience nothing else works in the beginning, until some lab culture is established, where the newbies obtain guidance by the experienced.
Of course this requires some set-up work in the beginning like creating some databases, setting up backup folders in print, organizing freezers, fridges and storage by assigning them coordinates etc. Some electronic lab notebooks (ELN) provide basic database capabilities for this and make lab life easier, but usually, they are quite expensive. Setting up an SOP might be a good start.
How you organize your storage (if you e.g. put all your primers in one set of boxes or store them together with the DNA samples of the respective projects or if you organize your fridge by the sample date) depends on your preferences. Probably some mixed mode is the most practical solution: Samples organized by projects, primers and master stocks organized in an individual storage and important things having at least one backup in a place that probably won't be destroyed in the same power or equipment failure/fire/earthquake/flooding/burglary.
HTH & good luck!
Wo