The MGB used a three synchromesh transmission up to 1968 and this unit is most likely to give trouble as it is less robust and spares parts are hard to find. On all MGB transmissions, noises and grinding are caused by worn bearings, gear teeth, or thrust washers. Also look for the gear lever jumping out of second or third gear when the clutch is not engaged. If the gearbox has problems it is best to replace with either a good second hand unit or a professionally recondition unit (with a warranty) as the cost of buying the parts yourself is usually more than an exchange item.
More than an inch or so of play in the clutch pedal indicates a worn master or slave cylinder that will require immediate replacement. Clutch replacement is best accomplished through the removal of the engine and gearbox.
Faulty overdrives may be suffering from a clogged filter, low gearbox oil level or electrical issues. If early overdrive transmissions fail to come in or out of third or fourth gear, a worn solenoid switch may be to blame. Check to see if the switch which prevents overdrive operation except in gears apart 3rd and 4th has not been shorted out as this will allow the overdrive unit to operate in reverse and will destroy the clutch arrangement in the overdrive unit.
Vibration especially at speeds over 50mph could be caused by worn universal joints at either end of the prop shaft or it could be the prop shaft itself which has gone out of balance. Universal joints are cheap and easy to repair but the prop shaft has to be removed.
The rear axle on an MGB can leak to due faulty seals, a clogged vent, or just careless overfilling. Never overfill the axle as the excess will contaminate the rear brakes. Clonking as the axle is engages is actually acceptable, but excessive noise will require attention to the rear differential although they will continue for some time in this condition.