car will not turn over when key turned

The most likely things that I can think of are a faulty ignition switch or a faulty relay in a fuse box.

It sounds to me like all the devices that are usually powered in position 2 of the ignition switch are not being powered. This could mean a faulty switch. The switch could however be powering a faulty relay, so check the relays first, but my guess would be the switch.

Does the radio switch correctly in position 1 of the ignition switch?

That the battery shows 13.6v when no current is being drawn from it doesn't prove the battery is OK. The battery may be damaged or old, and capable of showing good voltage (like 13.6) only with no load; when a load is applied, the voltage may instantaneously drop to, for instance, 8v. Your analysis and diagnosis will be more accurate if you first determine that the battery can provide the current required by the vehicle.

If the battery is in fact in good shape and fully charged, then:

The symptoms can also be explained by a bad connection in one or both of the big wires that run from the battery to the engine and chassis.

Loosen the connection at each end of both cables, remove the cables from the battery post or bolt, then clean the connector, the post, the bolt until the connecting surfaces are clean and shiny metal. Then remount the cables.

It's also possible (but less likely) that there's an internal fault in one of the cables or connectors, so that while the connecting surfaces are clean, electrical conductivity is broken somewhere invisible to the eye. If you have a charged battery and clean connections, and there's still no illumination of dash lights or the starter still isn't working, you'll have to use a voltmeter, or continuity tester, or rig a temporary test light to ascertain if one (or more) of the cables is faulty.