I don't know, I was told if I should move to Santa Monica this area is more lively and walkable than the rest. Are there any other areas that I should consider or is it hard to find apartments in all of Santa Monica ?
The reason you're told south of Montana is because north of Montana is rich folks homes, so that makes sense.
As far as walkability, it depends how much/often/far you plan on walking. The area you listed is all around the Promenade, which yeah, is nice walking area. But it's also pretty expensive.
Hilariously, they have recently build a ton of really nice income-restricted housing in the area, which means you have to make less than $60k or something, but the rent is also like $2k a month. So they expect you to take all your income and pretty much spend it on living there. No wonder they can't fill up the apartments (which Santa Monica is requiring them to do).
I'd say expand your search eastward in Santa Monica, above the 10, or just south of the 10 while still staying in Santa Monica. There are still a lot of pockets of nice walkability, especially if you're near Wilshire or Santa Monica Blvd. South of the 10 can get a little sketch though, so keep that in mind, even while they are improving that.
Santa Monica has a pretty nice city bus service, but if you want further transportation, I'd suggest looking in Culver City, where the Expo line just opened. This is expanding out to Santa Monica (to I think Colorado and 5th avenue), but it's not scheduled until 2015 or so, and I'm not sure how long you're planning on living in Santa Monica.
Anyway, that's some info. If you're just looking for walkability and not planning on traveling out of your home area too often, you might look at the Miracle Mile (WeHo) area. It sucks in that it's a hassle to get to/from wrt freeways, but there's some nice walking area there. I lived there for a few years and I could walk to the grocery, movies, restaurants, pretty much everything you could want, and did so often. Usually late at night, too.