Start out by getting a mason jar. You can buy them at pretty much any store, usually in bulk of 3 to 9. If you just want one, you can take a trip to a local antique store, where odds are they have loads of old canning jars for cheap. Mine is actually a family heirloom; my great-grandma used it to can vegetables during the Great Depression.
Decorate your jar. Glue things onto it. Do whatever you want to it to make it your own. Go crazy on it, or maybe even leave it plain. Paint it argyle or plaid or striped. Just make it you.
Start out your collection with a few small things. Maybe a movie ticket, or a friend's senior picture, or a cookie fortune.
Put your jar on a shelf where it's easy to get to, because odds are you won't want to keep up your collection if it's out of reach!
Every time you go anywhere and get some sort of little souvineer, a cookie fortune, maybe a perfume sample, a match book, a photo strip, perhaps even a candy wrapper, put it in your jar.
Final step? Go out and have wonderful times and crazy adventures that you can commemorate in years to come.