Learn Finnish: Marssimme hieman hitaammin. - We march slightly more slowly.

It can be either! For verbs ending in -ia like marssia (to march), the past tense marker -i- blends into the stem's existing -i-. This makes the present ('we march') and past ('we marched') look identical in the first person plural: marssimme. You have to rely on context to tell them apart.

How do we get hitaammin (more slowly) from the base word for slow?

The base adjective is hidas (slow). To form a comparative adverb like 'more slowly', you take the strong inflectional stem of the adjective, which is hitaa-, and add the comparative adverb suffix -mmin.

Could I use vähän instead of hieman?

Yes, absolutely. Both words mean 'a little' or 'slightly' and act as adverbs of degree here. Hieman sounds just slightly more formal or written, whereas vähän is extremely common in everyday spoken Finnish.