If the floor is not too dirty, a mop and warm water will be sufficient to clean it:
1. Use all-purpose cleaner dissolved into the water.
2. Prepare two buckets, one for the detergent, and the other for rinsing (Eventhough most of vinyl and linoleum manufacturer say that it is not necessary to rinse, it is better to rinse. One of the reason why these type of floors lost their shine is because they are usually not rinsed, and therefore gets dirty easily).
If the floor colour is still dull after rinsing, mop one more time with the mixture of vinegar and water.
Before you polish the floor by floor waxing, remove the old wax first. Use the mixture of three parts of water and one part of alcohol.

Scrub dirty part with nylon or soft steel wood pad which had been soaked into liquid detergent. Then mop the floor as usual.
Crayon could easily be removed by damp cloth and silver polisher.