Its best to start as the same race (or atleast dwarf & elf or man & hobbit) but you don't need to, once you've done the starter quests (level 5-6 i think) and get into the world proper, you can get insta travel between bree/shire/thorinshall/elf startingplace which puts you within 5mins of all the starting areas.
You can also run between them if you don't want to waste the money, as long as you stick to the roads and be a bit careful between the shire and bree you'll be okay.
I'm also pretty sure you can do most, if not all the epic prologue(?) quests that are separate for the different races until they merge around level 15-20.