The movie opens with a shot of Liam Neeson's car pulling up to a curb outside an electronics store.
 He gets out of his car and goes inside the shop, where he buys a karaoke machine
(for his daughter's birthday) from a guy wearing a turban.

Look closely at the street scene, and you'll notice a restaurant sign in the background which reads
"Musso & Frank" - the oldest restaurant in Hollywood. Liam is heading east on Hollywood Boulevard,
just a few yards east of Cherokee Ave (about a quarter mile east of Hollywood & Highland).

The scene inside the electronics store was filmed in a now-vacant storefront at
6630-B Hollywood Blvd, in Hollywood, CA, which is on the south side of the boulevard.
(There was once a store called "High Tech Electronics" at that address.)

The next L.A. scene, his daughter's birthday party, takes place at his ex-wife's mansion.
It's where he gives his daughter the karaoke machine (and her step-father gives her a horse).

You'll find that house at 56 Fremont Place, in the Hancock Park neighborhood of Los Angeles.

That's inside a gated community, at the southeast corner of Fremont and 8th Street.

Warning: This is a private home. Do not trespass on their property,
knock on their door, or do anything else that might disturb the residents.

Next, we see Liam picking up his developed photos (of his daughter's birthday) at a small store.

That scene was shot inside Valley Stores, at 11418 Moorpark Street, in North Hollywood, CA.

That's on the south side of Moorpark Street, at the southeast corner of Moorpark & Kraft Ave
(two blocks south of the Ventura Freeway and two blocks west of the Hollywood Freeway).

Liam agrees to come out of retirement for one day to act as a bodyguard to a pop singer during
a concert. When she is attacked by a crazed fan, Liam saves her life, earning her gratitude.

That scene opens with a night time aerial shot of the Staples Center (the round blue building).
You'll find the Staples Center on the south edge of downtown Los Angeles,
just east of the Harbor (110) Freeway, at 1111 S. Figueroa Street.

It is, of course, the home of the L.A. Lakers, and has been the scene of
quite a few superstar concerts over the years.

Next, we see Liam meet with his daughter at an upscale cafe. He is surprised when
her mother (his ex-wife) shows up as well. Both of them pressure him to give her
permission to go to Paris alone. Knowing the dangers, he is reluctant to do so.

This scene was shot half a mile east of the opening electronics store scene,
at the now-defunct Hollywood & Vine Restaurant, which was located on
the northeast corner of that famous intersection, at 6263 Hollywood Blvd.,
(just a few doors west of the Pantages Theatre).


                      

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Finally, we see Liam driving his daughter to the airport, where she will catch a plane to Paris
(and live to regret it). The airport is, of course, Los Angeles International Airport (LAX).

They are seen driving south on Sepulveda Boulevard, just south of the 96th Street bridge.

In the photo above, the camera is looking north/northeast up Sepulveda.
Here is a matching Google StreetView.

From there, the movie's action moves to Paris, and we don't see L.A. again
until the end of the movie, when the family is reunited at this same airport.