Generally a dispatcher (or TCO) should be trained in the following areas, although different handling agents and stations will have their own training programmes depending on customer and airport requirements.
Airside Driving Permit
Airbridge Licence
Triple AAA
Dangerous Goods
Fire Training
Ramp Safety
Manual Weight and Balance
GESAT (airport security training)
Unfortunately full computerised weight and balance training is no longer deemed necessary if loadsheets are produced by a CLC - the dispatcher simply has to check the loadsheet sent to them by CLC and hand it over, rather than actually produce the load plan and loadsheet themselves.
The extent of Manual W&B training given will depend on local requirements. Some airlines (for example most Middle East/US carriers) require dispatchers to be formally licenced in order to do a manual loadsheet, other airlines expect the flight deck to do their own manual loadsheets, and some expect the dispatcher to do a manual loadsheet if the need arises (eg system failure).
Lo-co's have reduced the need for weight and balance training to an absolute minimum by introducing standard loading rules and free seating. An easyJet TCO just fills in a simple loadchit with the passenger and bag figures, which the flight deck input into their laptop to calculate trim settings, ZFW etc. Ryanair also use an incredibly basic form of loadsheet. This enables handling agents to cut costs by reducing training and paying lower wages, but the downside is that IMHO these new lo-co style TCO's do not have the depth of knowledge that a fully load control trained dispatcher would have. They simply hand over a piece of paper to the flight deck...
It's still a great job, even if it is gradually being eroded... rumour has it Swissport are getting rid of dispatchers altogether for some contracts!