BACK: consonant changes with stem ___:
labial stops [pi, b, phi] usually become psi
This rule is universal in Greek--it applies to third-declension nouns, as well as to verbs (future and sigmatic aorist, but you will learn that not all Greek verbs form their futures and aorists with sigmas). When the final consonant of the stem is π, it's just a spelling rule: ψ = π + σ; when the final consonant is β or φ, a phonological change also occurs: the final consonant is devoiced (in the case of β) or deaspirated (in the case of φ), and then the spelling rule that replaces the devoiced or deaspirated consonant with ψ. (Similar principles come into play with the guttural series of consonants).
It's probably best not to think of the augment that applies to vowel-initial verbs ("temporal augment") as a contract or combination of epsilon with the initial vowel. As you will learn, there are phonological rules for vowel contractions that are different from the rules for temporal augment. And temporal augment doesn't behave consistently for each initial vowel: although there are discernible patterns, there are exceptions. The exceptions are not numerous, but you have to learn them verb by verb.
This is true of much of Greek verbal morphology: you have to learn principal parts for each verb, and although there are patterns, the principal parts of any given verb aren't necessarily predictable. That's what makes learning Greek so much fun. I'm sure you'll be delighted to learn that παύω, the verb that's often used as a paradigm, is just about the only thoroughly regular Greek verb. Even λύω undergoes a shortening of the long υ in certain forms.
You can make some sense of the vast ocean of irregularities that seems to characterize Greek verbs with a little knowledge about the history of the language. That's probably too much for you at this stage, but eventually, I think it's helpful to put Greek verbs in perspective.
The -mi verbs will require learning principal parts verb by verb. Sorry, but there's no way around it. Fortunately there aren't many -mi verbs, though most of them are very common. The only productive classes are those in -υμι, and these verbs are relatively regular once you get past the present/imperfect.