Cheryl C.
Yelp
Pretty much all of the city centre located, or nearby galleries in Edinburgh are really impressive in terms of architecture, and a grand joy to observe, which is a good starting point, considering this is just the impression you get before you even see any of the art.
I'm really fortunate to have both the Dean Gallery and The Scottish National Gallery of Modern Art close to where I live, and in fact, if you're in the centre of town, The Dean is just a short walk from the west end, and a pretty nice walk at that, particularly on a sunny day (we do get them sometimes).
The Dean houses the Gallery of Modern Art's permanent collection, and also exhibits its own temporary shows. The permanent collection includes Dada, surrealist art and the amazingly intimidating Vulcan sculpture by Eduardo Paolozzi, which is well worth a trip to see. There is also The Paolozzi Gallery, which includes more work from the late Edinburgh born artist and sculptor. I recently saw the Diane Arbus photography exhibition here, which I thought was compelling and excellently displayed. I strongly recommend taking a wander away from town to get here, because after you've finished at Dean Gallery, there is the added luxury of being able to simply cross the road and have a peek, or more, at the Gallery of Modern Art.