As a local, i tend not to rent cars from Edinburgh so my comments regard what i think the ease must be like for those that do! (clumsy way of saying that non-local knowledge is probably more relevant here!).
The airport is outside the city obviously so my assumption is that it will be easier for you to start your journey from there given that you've not got to navigate out of the city. getting out to the airport is very very easy and the airport buses are cheap and quick. once at the airport there is plenty competition from avis, hertz, national, dollar, etc... i'm sure you'll find your preferred renter of choice there.
the case for the city centre is less clear: what i do know from colleagues is that the choices of cars available are poor since they don't keep a large stock of cars in the city centre, compared to the airport and secondly not all the competing rental firms are located together and are actually spread about from the train station, to omni centre, to haymarket and murrayfield being the larger ones.
driving in edinburgh is fairly easy - parking is not. there are speed cameras galore and green lanes to avoid. i'd encourage you to consider using the airport as your starting point.
the poster commenting on petrol (gas) stations in the vicinity is absolutely right that they are not obvious but they do exist about a mile either direction. if coming in from the West, the roundabout that connects the M8 with the M9 is the best and if coming from the East, at the Gyle roundabout, turn into the shopping centre and there's one there. i can think of about another 2 or 3 that are not that much further away, but these ones are the most obvious and closest. one of the aspects i've always liked about US airport rental locations is that there always seems to be a nearby gas station within sight of the rental offices! sadly, Edinburgh isn't so blessed.
hope that helps - i think however you're making the right choice in renting a car since whilst the public transport system is actually very good, you will benefit from the flexibility and freedom of having your own wheels.
happy hunting!