Learn Russian: Я каждый день хожу в школу.

The phrase каждый день (every day) tells us this is a repeated, habitual action. For regular round trips like going to school and coming back, Russian uses the multidirectional verb ходить (conjugated here as хожу), rather than the one-way verb идти.

Since this lesson also covers ездить (to go by transport), could I use езжу instead of хожу?

You would only use езжу if you specifically want to emphasize that you commute to school by car, bus, or train. By default, Russians use the on-foot verb ходить to talk about attending an institution like a school or university on a regular basis.

Why is the word for school written as в школу instead of в школа?

When you talk about motion towards a destination, the preposition в (to/into) requires the Accusative case. For feminine nouns ending in -a like школа, the ending changes to -у.

Why don't the words каждый день change their endings?

They actually are in the Accusative case! Time expressions telling how often something happens usually take the Accusative. However, because день (day) is a masculine inanimate (non-living) noun, its Accusative form looks exactly the same as its basic dictionary form.