I've read that the most commonly used verbs in a language are almost always irregular, and for the most part the irregular forms of verbs can be traced to an archaic form being preserved in the language, or to the influence of other languages. When creating a naturalistic conlang, what verbs should be irregular? I'm already aware that 'to be' is almost always irregular, but are there any general patterns for which verbs are most often irregular in a natlang? (For example, are verbs involving movement more commonly irregular than not?)
The more used a verb is, the more likely it is to resist evolution. So the most popular verbs are likely to be the most odd. Think of the activities which were more common during the evolution of the language. A conlang spoken by a prosperous society might have more irregular verbs related to possession, social interactions and art - while a conlang spoken by nomads living in a land with cataclysmic climate would have more irregular verbs about natural phenomenons and actions that everyone has to perform in order to guarantee their survival.
Note that verbs can have varying degrees of regularity. English has a pretty clean system, with a few boldly irregular verbs, a decent list of somewhat irregular verbs and a lot of very regular verbs.
Meanwhile, French verbs tend to be split across different groups based on their ending. There's -er, -ir, -oir and -re. Verbs within one group tend to have similar conjugations. Still, there is so much irregularity in French verbs that students often refer to verb conjugation volumes that have the full conjugation of about 100 model verbs. At the end of those volumes is a list of all French verbs with the page number of the most similar model verb.
EDIT: scrolling through the list of model verbs in a Bescherelles conjugation volume, those are the verbs which seem less regular to me (native French speaker):
- to be - être (also an auxiliary)
- to have - avoir (also an auxiliary)
- to go - aller (sometimes used as an auxiliary)
- to die - mourrir
- to know - savoir
- must - falloir (it's a whole verb in French)
- can - pouvoir (it's a whole verb in French)
- to have a duty [to do something] - devoir (mostly used as an auxiliary)
- to be of value - valoir
- to want - vouloir
- to rain - pleuvoir
- to drink - boire
- to do / to make - faire (not an auxiliary in French)
The first class of verbs that is often highly irregular are the auxiliary verbs. This does not only comprise to be and to have but also the modal auxiliaries (like must, can, shall, and will). They are used very frequently and tend to erode phonetically, and they are also prone to suppletion (showing a mixture of different stems from originally different words).
A quick check (for my native language German) seems to confirm your intuition that verbs of motion are more often irregular than average. They are also used quite frequently. In Romance languages, the basic verb for to go (e.g., French aller or Italian andare) is very irregular and mixing several different stems ((vulgar) Latin ambulare/allare, vadere, ire).
Some verbs are irregular because they rhyme with an irregular verb, e.g., German schreiben, schrieb, geschrieben "to write", borrowed from Latin scribere, became irregular because of rhyming verbs like bleiben "to stay".
Well, the statement is demonstrably untrue. Chinese (Mandarin and Cantonese) doesn't have irregular verbs, because verbs aren't conjugated. Quechua doesn't have any. And languages like Inuktitut has no irregularity in verbs because of the way that the language assembles words based on a root+affixes.