歩く vs. 歩いていく (aruku vs. aruite iku) + other forms

You use 歩く (aruku) when you are referring to any of the following: (1) act of walking, (2) your destination, or (3) where you are actually walking:

あの人は歩くのが速いですね。
Ano hito wa aruku no ga hayai desu ne.
That person walks fast, doesn't he?

学校まで歩く。
Gakkou made aruku.
I walk to school.

あそこを歩いている人はだれでしょう。
Asoko wo aruite iru hito wa dare deshou.
I wonder who walks over there.

Note the particles used for  歩く : まで and を. If one wants to use に or へ, then 歩いていく or  歩いてくる would be the proper forms used:

学校に歩いていきます。
Gakkou ni aruite ikimasu.
I go to school on foot.

When one talks about walking up an incline, the appropriate verb is  歩いて登る (aruite noboru) or simply 登る ("climb"). The 歩いて before 登る can be thought of as an adverb meaning "on foot".

階段を歩いて登る/登る。
Kaidan wo aruite noboru / noboru.
I walk up / climb the stairs.

When one talks about taking a stroll, the correct verb is NOT 歩く, but rather, 散歩する (sanpo suru).