You can enable Serial Numbers by clicking on Settings > Account Setup. Tick the box next to "Enable Serial Numbers" and further options will expand.
You can enable them when shipping (i.e. when selling and receiving Finished Goods) or when shipping and receiving (i.e. also when consuming Raw Materials and shipping goods as well).
Note that if you want to make sure that no Serial Number is the same with another Serial Number across your entire product range you should select 'Serial number to be unique across all products'.