It is all about the size*
*no that is not what she said.
As space travel happens all relative to gravitational sources, travelling from A to B in space can be achieved by something as fancy as a 2-story-house and a bunch of fire extinguishers.
Or by a highly specialized racing ship with huge fusion drives and fancy radiator fins to get read of the heat before it cooks the pilot(s).
Although no matter which of both representations fits your racing ships better, they still end up having to traverse huge distances while being in motion themselves.
So how would you lay out a race track in space?
Around the World
As you already mentioned, there is the around the world format where you set certain checkpoints which must be reached sequentially. This could be achieved by placing these checkpoints in stationary orbits around a planet and have the racers pass each of them.
Which would end up in a contest of navigational computers, the more money you got the better your racing skill. Alas you could counteract this by introducing certain rules to racing, e.g. no computer assistance, which would mean physicists would become the best racers - or every racer becomes a physicist.
Now, described above is an easy racing track, as every checkpoint is static relative to other checkpoints.
If you wanted to spice the whole thing up a little (and introduce some nasty headaches) you could put checkpoint at different orbital planes around a gravity source, effectively making them move.
Around the 'Verse
It's space, and there is literally a lot of it. So why not make them race from system to system, making certain planets or space stations checkpoints? In the end you'd still end up with most of the racing being navigation and controlled translation burns. But that's how it works.
How would racing work?
Now as you will know, the faster an object is going, the less time it takes for it to move from point A to point B. In space these distances and the speeds involved to travel between these distance are a multiple of those on earth. Also there is no friction in space, which means the only reliable break system is to rely on getting caught in the gravity well of a small object (e.g. passing near a planet in order to have it's gravity pull you off course - the base assumption underlying gravity slingshot maneuvers) and retro burning, essentially the practice of decelerating by firing your thrusters against your current direction of motion.
Making the fear and thrill of overshooting your target by millions of kilometers the most exciting thing in the process of racing.