Just a few observations:
ἑάφθην is an obscure word that occurs only in two places in the Iliad (except for scholia and glosses attempting to explain it). There's a separate entry for it in LSJ, which explains it as equivalent to the aorist passive of ἅπτω, ἥφθη, but there are other explanations, and it's not even clear whether it begins with a rough or a smooth breathing. (Incidentally, the abbreviation v. infr.
in the LSJ entry for ἅπτω doesn't relate to ἑάφθην; it's an abbreviation for Latin vide infra
("see below") and it's an instruction to look below in the entry for the middle forms ("Med.
" for Latin medium
), which are indeed in the next section of the entry.)http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3De%28a%2Ffqh
But the form ἥφθην, although listed in Smyth, doesn't seem to be attested anywhere--there are no citations for this form in LSJ under ἅπτω. It might be inferred from the future passive form συναφθήσομαι, which occurs at least once in the 2d century CE medical writer Galen, but LSJ does give any indication that ἥφθην appears in any ancient Greek text.
In addition, the bracketed form ἧφα doesn't seem to occur anywhere, either, and it's possible that the stem underlying ἧμμαι was ἧπα, but there's no way to tell--although a number of verbs with roots in -π form their perfect stems with -φ-, e.g. βλάπτω.
So really there are only four securely attested principal parts for ἅπτω: ἅπτω, ἅψω, ἧψα, ἧμμαι.http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Da%28%2Fptw
Learning principal parts is easier and works better for Latin because the Latin verb system was much more radically restructured than Greek, which preserves lots of archaic features without analogical levelling. Even the supposedly irregular perfects and past particples of 3rd and 4th conjugation Latin verbs can be explained by a few simple morphophonemic rules. Latin principal parts, even nearly all of them that seem irregular at first blush, group into distinct and easily assimilated patterns. (Latin was also much more standardized in the classical period--in contrast to Greek, where there are many divergent dialects to contend with.)
But Greek is different: the morphology is much more complex, with thematic and athematic presents and aorists, first aorists and second aorists and perfects, reduplication, apophony (vowel mutations such as λείπω-ἔλιπον-λέλοιπα, but we still have these in English, too!), futures that take middle forms, etc., as well as more suppletive verbs (φέρω-οἴσω-ἤνεγκα, but in English, we have "go", "went", "gone"; "be", "am" "is", "was").
Here again, some morphophonemic rules are helpful to understanding how various stems are formed from a single root, but at an elementary stage, explaining these rules would probably be more confusing than helpful. So brute memorization is a necessity. There are very few "regular" verbs in Greek.
If you want to understand how the many ways that various stems (present-aorist-perfect) and other forms of Greek verbs are formed from a single root, it's worth working through the entire discussion of verbs in Smyth. I think that understanding nasal infixes, present reduplication in -i- and the behavior of the suffix -y- in various phonological contexts (which is not usually taught) are particularly helpful for isolating the verbal root on which the various stems are formed. This won't allow you to predict the various stems on which the principal parts are based, because for each stem there are several alternative procedures. But understanding the underlying processes will help in memorizing the principal parts and, more importantly, in retaining them once memorized.
But ultimately, my advice would be: if you can't stand brute memorization, don't study Greek.