"Be-" is a very old prefix, and often the meaning of the verbs with and without that prefix has changed over time. Sometimes it's just used to make the verb transitive ("zahlen", "etwas bezahlen"), sometimes it makes the meaning more abstract ("auf etwas mit dem Finger deuten", "etwas (im Geist) bedeuten" are actually pretty different in daily use), and sometimes it intensifies.
You can explain "suchen" (search for s.th., seek s.th., move toward s.th.) and "jemanden besuchen", "jemanden aufsuchen" (seek s.o., move toward s.o., visit s.o.) in such a way.
For learning, it's better to try not to attach a particular meaning to be-, ent-, ver-, zer-, but instead learn the verbs as words with their own separate meaning.