Spanish doesn't have a distinct class of modal verbs. That is a typically English (or rather Germanic) grammatical feature. English modal verbs are characterized by a defective conjugation ("I can, you can, he can", not "he *cans"); nothing like this happens in Spanish, where verbs like poder and deber are conjugated normally, albeit with some irregularities.
On the other hand, verbal periphrases are found also in English ("be going to" is one of them); these do have parallels in Spanish. "Periphrasis" just means "turn of phrase", in linguistics, "expressing a simple concept using a whole phrase". So English passive voice is periphrastic too (the verb "to be" plus the past participle), and Spanish has both a periphrastic passive voice and another type (the so called reflexive passive or pasiva refleja).
Auxiliaries are also found in both languages, but Spanish only uses one, the verb haber, for the compound tenses, while English uses have for the compound tenses and also will for the future (one could argue that would is an auxiliary too; "auxiliary" and "modal" are not mutually exclusive).
Spanish has a smaller range of "modal" verbs (basically poder and deber), but all verbs including these two can appear in the subjunctive mood, or they can be conditional (roughly an equivalent of using English would). So what English expresses using modal verbs, Spanish shows by using poder, deber, some equivalent turns of phrase that refer to ability and obligation, and the contrast between the conditional, the plain indicative and the subjunctive. You have to study it without reference to English, because the two systems don't match, although of course translation is usually possible.
In general the English -ould auxiliary/modals translate to conditional verb conjugation in Spanish:
- "he does" = hace
- "he would do" = haría
- "he could do" = podría hacer
- "he should do" = debería hacer
The subjunctive is too complex a topic to deal with in a general question. The old English subjunctive is largely moribund ("he requested that she be buried next to him if she were to die before him") and Spanish subjunctive is applicable to many kinds of statements.