Formation of perfectum is pretty complicated. Most handbooks advise to memorize this form of every verb, but it's not an absolute necessity.
Most popular endings are -vi and -ui :
I conjugation -are > -avi, IV conjugation -ire > -ivi.
In II and III conjugation it's -ui, but especially in the III conjugation most verbs form perfectum irregularly.

I conjugation: Most verbs are regular here, the ending is -avi (laudare, laudo, laudavi)
Exceptions :
1. Two frequent verbs with reduplication:
dare > dedi , stare > steti, its compounds : constare > constiti, praestare > praestiti
2. In one verb, there is a change -are into -i, with vowel lengthening: iuvare > iuvi
3. Verbs changing -are to -ui : sonare (sonui), vetare (vetui)

II conjugation: The regular ending is -ui, but many verbs are irregular (habere, habeo, habui)
1. Change of -ere to -i, mostly with vowel lengthening:
videre > vidi, movere > movi, sedere > sedi, cavere > cavi, solvere > solvi, respondere > respondi
2. Some verbs take -ss- or -s- : iubere > iussi, manere > mansi, ardere > arsi, augere > auxi, ridere > risi
3. Some verbs change -ere to -evi : flere > flevi, implere > implevi
4. Two verbs become deponentia in this tense : audere > ausus sum, gaudere > gavisus sum

III conjugation: Here most forms are irregular. Regular ending is -ui.
Frequent verbs having -ui are : colere, deserere, consulere, alere, rapere, eripere
Also verbs in -uere have -ui : metuere > metui, constituere , statuere, instituere

1. Some verbs change the consonant before -ui : ponere > posui, imponere (imposui), componere, proponere
Also : gignere > genui

2. Very many verbs have -s- or -x- :
dicere > dixi, mittere > misi (amittere, committere, permittere, promittere, dimittere), ducere > duxi (deducere, adducere, educere), vivere > vixi, trahere > traxi, gerere > gessi, premere > pressi, cedere > cessi (concedere, discedere, procedere, recedere), tegere > texi, claudere > clausi, sumere > sumpsi (consumere), accedere > accessi, contemnere > contempsi, dividere > divisi, fingere > finxi, cingere > cinxi, intellegere > intellexi, iungere > iunxi, spargere > sparsi, regere > rexi, scribere > scripsi, vehere > vexi, fluere > fluxi, laedere > laesi, edicere > edixi
More irregular are: surgere > surrexi, pergere > perrexi, diligere > dilexi ;
aspicere > aspexi, respicere > respexi

3. Change of vowel, mostly a into e, i into e :
agere > egi
facere > feci (efficere > effeci, deficere, interficere, conficere, afficere)
capere > cepi (accipere > accepi, recipere, excipere, incipere, suscipere, praecipere)
cogere > coegi, exigere > exegi, colligere > collegi, eligere > elegi, iacere > ieci
With consonant change : frangere > fregi.

4. Change of -ere to -i, mostly with vowel lengthening:
with loss of -n- or -m-:
vincere > vici, relinquere > reliqui,rumpere > rupi (corrumpere > corrupi), contingere > contigi, fundere > fudi , effundere > effudi
without additional changes :
legere > legi, fugere > fugi, vertere > verti (revertere), ostendere > ostendi, defendere > defendi, accidere > accidi, occidere > occidi (incidere), descendere > descendi, convertere, advertere, intendere, ascendere

5. Ending -ivi: petere > petivi (repetere), quaerere > quaesivi, cognoscere > cognovi, noscere > novi, crescere > crevi, cernere > crevi (decernere), sinere > sivi, quiescere > quievi, cupere > cupivi
Ending -ii : desinere > desii.

6. Reduplication :
Verbs in -dere > -didi : credere > credidi, reddere > reddidi, tradere > tradidi, addere, perdere, condere, edere, prodere
Other cases : consistere > constiti, cadere > cecidi, canere > cecini, parcere > peperci, discere > didici, poscere > poposci, fallere > fefelli, tendere > tetendi, tangere > tetigi, currere > cucurri (occurrere), pellere > pepuli, caedere > cecidi, pendere > pependi, parere > peperi

IV conjugation : The regular ending is -ivi
Exceptions :
1. long vowel : venire > veni (and all verbs in -venire)
2. sentire > sensi, reperire > repperi, aperire > aperui

Irregular verbs: esse, velle, ferre, ire, fieri, tollere
esse > fui, posse > potui,
with prefixes: deesse > defui, superesse > superfui, interesse > interfui
with change of prefix: adesse > affui, prodesse > profui, abesse > afui

ferre > tuli, conferre > contuli , inferre > intuli , deferre > detuli
referre > rettuli, afferre > attuli, auferre > abstuli, offerre > obstuli, differre > distuli

velle > volui, nolle > nolui, malle > malui

ire > ivi or ii, most verbs with prefixes have -ire > -ii : perire > perii, redire, transire, abire, edire, exire
fieri > factus sum, tollere > sustuli

No perfectum : inquam, aio
Only in perfectum (no present forms): coepi, odi, memini