Geocaching sans GPS
Speaking as someone who did this before(14 caches), for those of us who want to use a cache listing service, it's a bit of work to find a cache.
First, find the cache you want online. Memorize or print out the description (or at least the important bits), read the hint, and commit it to memory (I used sticky notes for hints, actually). Go to the mapping website of your choice, and note any intersections or features the cache is nearby (such as creeks) or just print out a map. Once you get that done, it's time to find the cache.
Arriving at the site, you usually have no idea where access points for the cache are. You will have to look around a bit, then once you find a trail that looks promising, try to keep track of where you are on the map.
Once you think you are within 30 feet of the cache, start searching. Look in places appropriate for the cache container (You DID note the cache size, right?), and hope you find it. You may make mistakes as I did, and find a water bottle, only to discover it's filled with beer or urine. If, however, you find the cache, you should be proud of your accomplishment. Sometimes it will take multiple trips to find the cache, or sometimes you'll get lucky.
Happy hunting, and good luck!
Another option, posted by a different user:
The cache density in some locations is so great that you can find caches not only without benefit of GPS, but also without benefit of a cache listing service. Stroll through just about any small city park, try to think where you might hide a cache, then keep an eye out as you pass through that area. Chances are good that, sooner or later, you will find some Tupperware or ammo box under the shrubs just off the path.