There are several different launch scenarios to put a satellite into GEO. Some satellites are launched directly into GTO and stay there for perhaps a few days until the geometry and everything checks out. Others fire their apogee kick motor on the first apogee. It depends on the particular mission design. The amount of radiation the satellites receive by passing fairly quickly through the radiation belts isn't excessive, especially for satellites designed to operate above the belts.

Most satellites avoid spending their entire operational life within one of the radiation belts but there are some that do. The GPS satellites are in near circular orbits at about 20,000 km which puts them in the upper belt. There are bound to be some others (Gorizont, scientific satellites, etc.) There are some satellites, such as the Russian Molynia and LDS series, that have highly eccentric orbits that cross both van Allen belts twice every 24 hours (12 hour semisynch orbits) and do so for months or even years. Those satellites were designed to survive under those conditions and do so remarkably well.

Manned missions don't venture into the van Allen belts as a rule. Gemini 11 had an apogee of 1374.1 km for a while, a record for manned orbital apogee that still stands. That orbit took it about 600 km into the lower belt. All of the Apollo moon missions crossed the belts very quickly so the radiation exposure was limited. Orbiting humans within or above the van Allen belts would mean we'd need to accept the additional mass of shielding and/or higher radiation exposures.