There is no real way to predict the perfect stem or the supine stem (past-participle stem) of a Latin verb; there are only probabilities. One normally learns the past stems of a verb along with its present stem and conjugation group if they are irregular.
The regular suffix to form the perfect stem is by adding -v- to the present stem, so after the theme vowel. This is very common, especially for verbs of the -a- conjugation, and also somewhat common for the -i- conjugation.
voca-re → voca-v-i
audi-re → audi-v-i
quaer-ĕ-re / quaes-ĕ-re → quaes-i-v-i or quaes-i-i (r changes into s)
Another common way of forming the perfect stem is by removing the theme vowel (if present) and adding -u-. This is especially common for verbs of the -ē- conjugation, but by no means limited to it.
terrē-re → terr-u-i
evanesc-ĕ-re → evan-u-i (ingressive present suffix -sc- disappears)
Adding /s/ is also quite frequent, especially with consonant stems. The resulting letter may be an x if the present stem ended on -c- or -g-.
mitt-ĕ-re → mis-i
teg-ĕ-re → tex-i
torquē-re → torsi
Many verbs do not get a suffix at all, especially but not exclusively with consonant stems, but the vowel in the stem will be lengthened compensatorily, and sometimes changed qualitatively:
vĕni-re → vēn-i
vĭdē-re → vīd-i
vinc-ĕ-re → vīc-i (nasal present infix disappears)
fund-ĕ-re → fūd-i (nasal present infix disappears)
ăg-ĕ-re → ēg-i (vowel changes)
făc-ĕ-re → fēc-i (vowel changes)
Many verbs of the -i- conjugation have both -v-i and -i:
audi-re → audi-i / audi-v-i
i-re → i-i / ī-v-i
Some verbs undergo reduplication, which is an artefact from proto- or pre-Latin, if I remember correctly. This, too, is most common with consonant stems. The vowel will often change in length and quality.
fall-ĕ-re → fe-fell-i (vowel changes)
caed-ĕ-re → cĕ-cīd-i (vowel changes)
tang-ĕ-re → te-tig-i (nasal infix disappears; vowel changes)
făc-ĕ-re → archaic fe-fec-i (vowel changes)
păr-ĕ/i-re → pe-pĕr-i (vowel changes)
pell-ĕ-re → pĕ-pŭl-i (vowel changes)
posc-ĕ-re → po-posc-i (o-reduplication)
spondē-re → spo-pond-i (apparently unique, based on Proto-Indo-European perfect)
But none of these rules are remotely strict or consistently applied. And many verbs have several perfect stems that can be used interchangeably.
The supine stem (past participle) is regularly (but by no means always) formed by adding -t- after the theme vowel, if present, where theme vowel -ē- changes into -i-. This -i-tus sometimes also happens with consonant stems. Supine stems are just as irregular and unpredictable as perfect stems, and many verbs even have two interchangeable supine stems.
voca-re → voca-t-us
audi-re → audi-t-us
terrē-re → terr-i-t-us
pet-ĕ-re → pet-i-tus
cognosc-ĕ-re → cogn-i-t-us (ingressive suffix -sc- disappears)
quaer-ĕ-re → quaes-i-tus (r changes into s)
However, in many cases the suffix -t- has to come immediately after a consonant; in that case, many things can happen. This is perhaps unexpectedly not limited to consonant stems.
After p and c, it can come immediately after:
cap-ĕ-re → cap-t-us
fac-ĕ-re → fac-t-us
The dental consonant -t- plus -t- often but not always becomes -ss-. This sometimes also happens after -d-.
mitt-ĕ-re → mis-s-us
ced-ĕ-re → ces-s-us
But -d- plus -t- usually becomes -s-.
fund-ĕ-re → fus-us (nasal present infix disappears)
caed-ĕ-re → caes-us
accend-ĕ-re → accens-us
spondē-re → spons-us
After -l-, it is turned into -s-.
fall-ĕ-re → fal-s-us
pell-ĕ-re → pul-s-us (vowel changes)
Consonant g becomes unvoiced:
ag-ĕ-re → ac-t-us
frang-ĕ-re → frac-t-us (nasal infix disappears)
Consonant b becomes unvoiced and t may or may not turn into s (but verb stems on -b- are rare at any rate):
lab-i (deponens) → lap-s-us
scrib-ĕ-re → scrip-t-us