Before you read on, be warned: this can lead you down the path where at the end you will be remembered as the person who brought down Prod. I'm assuming you don't want to do that. So be very careful.
There is a class that can help you with server side caching: GlideCacheManager. In order to cache something, first one needs to be created by calling its addPrivateCacheable method:
GlideCacheManager.addPrivateCacheable('<cache name>');
Afte that you can add key-value pairs to the cache:
GlideCacheManager.put('<cache name>', '<key>', '<value>');
To read a cached value:
GlideCacheManager.get('<cache name>', '<key>');
To remove a key-value from cache:
GlideCacheManager.prefixFlush('<cache name>', '<key>');
To clean up the cache:
GlideCacheManager.flush('<cache name>');
And here you need to be very careful: always check that the cache already exist before putting something in there. Don't overdo the quantity of the data added to cache. Keep some last usage time or something in every cache you create (but create as few caches as possible) and clean up caches not used for a while using some scheduled process. I would also be careful to only place stuff in cache that are not meant for everybody do see (like HR data) - the private cache might be read by someone unauthorized.
Again, it is a powerful tool, but can get you hurt.
As for a client side, you could use standard web technologies, like SharedWorker. Set one up, load all scripts needed and communicate with it from any page of the same site.
