Personally, I loved cruising around tracks solo in time attack, but just free roaming, and taking in the scenery. You can spot some really neat things if you just stop and take the time to look around the tracks (Or try to get to places where you're not meant to.
Some of my personal favourites are:
Peach's castle in Royal raceway in MK64 (It's just loads of fun to look around, and certainly makes an interesting idea for an improvised battle course.)
Sign browsing (Some of the signs in Mario Kart are really entertaining. My favourites are the "Koopa Kart", and "SHERBERT". Ones.
Also, you can't forget the signs in Koopa City. Bowser rocks!)
Driving the minecart tracks backward in Wario's Gold Mine on Mario Kart Wii. (Ledge-hanging to dodge those carts as they head your way as you make your way backwards was a lot
of fun. :L)
...and finally, the leafless tree in MK7's Alpine Pass (I know that this is obviously not intended, but on the Alpine Pass track, there is a tree just off the track's main path that has no top. It is literally a vertical log. Indeed a neat spot for the avid Mario Kart sight-seer.