Learn Croatian: Što mi savjetujete kao brzo rješenje? - What do you advise me as a quick solution?

The verb savjetovati (to advise) requires the dative case for the person receiving the advice. mi is the short dative form of ja (I), meaning 'to me'. If you used the accusative mene, it would sound like you are the person being recommended to someone else!

How does the dictionary verb savjetovati become savjetujete?

For Croatian verbs ending in -ovati, the -ova- part changes to -uj- in the present tense. First you find the base savjetuj-, and then you add the formal or plural 'you' ending to get savjetujete.

What case is brzo rješenje in after the word kao?

It is in the accusative case. The word kao (as / like) doesn't trigger a specific case on its own; it just connects things. Here it links to the direct object of the sentence (what is being advised). Because rješenje is a neuter noun, its accusative form looks exactly the same as its nominative (dictionary) form.

Since this lesson is about trebati (should / to need), how could I ask this using that verb instead?