The big dance contest (and some of the interior classroom scenes) were shot in East L.A., in the gym at
Huntington Park High School at 6020 Miles Ave, in Huntington Park (about 14 miles east of Venice High).

      

After being dumped by Sandy, Danny gets jealous when he sees her with a high school jock, so he gets
desperate enough to try out for both Rydell High's baseball team and its basketball team. He doesn't make it.
(But later, he manages to win his letterman jacket in track.)

These athletic try-out scenes were also shot at Venice High School, at 13000 Venice Blvd., in Venice, CA.
The baseball diamond is located behind (south of) the school's main building, just to the east of the track,
while the basketball courts are located on the far east side of the campus.

The big car race, near the end of "Grease", was filmed on the dry riverbed of
the Los Angeles river, on the east side of downtown L.A.

The starting line (at the south) was under the 6th Street bridge
(its base can be seen in the foreground of the above and below photos).

Heading north, the speeding cars passed under the 4th Street bridge (seen in the distance above), and
kept going. After finally passing the 1st Street bridge (seen in the photo below), they U-turned and headed back

In the shot below, we are looking south, as the cars speed up the embankment of the riverbed,
heading back towards the 6th Street bridge (which you can see in the distance below).

.

(Warning: This part of the Los Angeles River is in a rough part of town. Exercise reasonable caution.)

The big school carnival scene at the end (where John and Olivia performed You're the One that I Want
and the whole cast performed We'll Always Be Together) wasn't filmed at Venice High.

(You can tell that by the mountains in the background...
in one scene, you can actually see the Griffith Park Observatory.)

These carnival scenes were actually shot at John Marshall High School,
which is located at 3939 Tracy St., in the Los Feliz district of Los Angeles.

(In the scene above, the camera is looking northwest.  In the scene below, the camera is looking south.)

They set up the carnival rides in the center of the school's track/athletic field,
which borders Griffith Park Blvd, in back of (northeast of) the school's main building.

(In the scene below, the camera is looking northwest.)

Besides "Grease", other films shot at John Marshall High include
"Pretty in Pink", "Evan Almighty", "Zapped", "True Colors", interiors for "Rebel Without a Cause",
and the TV series "Growing Pains" (where it was called "Thomas Dewey High").