Working just with the basic game (read: PHB, no expansions), I was trying to outfit an elven cleric, but the starting equipment only includes simple weapons (or war hammer, with which dwarves are proficient, but not elves). A high-dexterity elf could really benefit from access to longbow or short sword, but neither one is available as standard equipment for a cleric at character creation. How do I get elf-appropriate starting equipment that won't limit my damage potential?
This seems to be a broader problem for other classes as well. Indeed, every class that allows you to equip martial weapons at the start also grants martial weapon abilities — making the elf weapon training redundant. Did I miss some special rule that allows you to treat "race-specific weapons" as simple weapons for initial character creation? I'd even be happy if shortsword was "simple", but it isn't.
Best Answer
Elf weapon training can be used for classes with no weapon Proficiencies
When the class already has martial weapon Proficiencies, this is redundant, but it allows other classes, such as sorcerers, for example, to make good use of weapons. This would be mostly useful for the longbow, less so for the shortsword, since these characters do not want to get into melee, and even less for the longsword since it uses strength.
Another case where this is very interesting would be a combat oriented bard (as opposed mostly magic oriented): Such a character would heavily invest in DEX but does not get a longbow proficiency. Supplementing that as elf seems like a good option.
You can use starting wealth to buy equipment
The starting equipment for any class is adapted to the class and does not consider the race. If you want to choose your equipment freely, buying equipment would be a good idea. Otherwise, it is always possible to ask the GM if you can exchange a war hammer for a shortsword, even if it is not possible RAW.