Geocaches are listed on one or more of several web sites.
geocaching.com is by far the largest site worldwide. It is a commercial site run by Groundspeak, Inc. from Seattle in the U.S.A It is well organised and profesionally run. Anyone can sign up for free, to try it out, but as a Basic Member there are restrictions on the facilities that you can use and the caches that you can see. Anyone who takes up the game seriously really needs to subscribe as a Premium Member.
Opencaching UK is a very much smaller site, with far fewer caches listed. Such coverage as there is, is patchy. But the site is non-commercial, and the full range of facilities is free to all. Being non-commercial frees the site from some restrictions and anomalies. Being so much smaller allows it to be more responsive to users, more flexible and more innovative. But I must declare an interest, as I am the primary admin for opencache.uk.
There are too few caches currently listed on opencache.uk for it to be the only site an active geocacher uses. While opencache.uk would welcome more caches, it sees little benefit in listing caches that are already listed elsewhere. Its aim is to focus on quality rather than quantity.
There are opencache sites in many countries, such as Germany, Poland, Romania, The Netherlands, Australia, U.S.A. etc.
TerraCaching is a smaller U.S. site, but its use in the U.K. has declined in recent years.
You can use a dedicated GPS receiver for caching, and these work well with all the sites. A modern smartphone, for which there are several apps, has both advantages and disadvantages compared to a dedicated unit. Most of the apps for smartphones focus on geocaching.com, but c:geo on the Android platform supports opencache.uk equally.