I assume you're talking about the 300/4L IS lens since the 300/2.8L IS doesn't take a front filter.
You don't really need a protective filter, but if it makes you feel more confortable to use one you'd need a 77mm multicoated UV filter.
I'd recommend one of the HOYA line, probably something like the
Hoya PRO 1 Digital UV filter.
What you don't want is a cheap filter that will degrade image resolution or an uncoated or monocoated filter which will lower image contrast. Good 77mm multicoated filters will cost somewhere between $50 and $100 or so. I'd beware of putting anything cheaper than that on a good lens like the Canon 300mm.
Hoya filters are generally good, but avoid the low cost "green box" Hoya filters (so called because they come in a green box!) since they are only monocoated and can increase flare and decrease contrast at times.