I know "Truth, Beauty, Goodness" is "Veritas, Bonitas, Pulcritudo."
But do I need an "et" before "Pulcritudo"? When do you use and's in Latin?
And how would convey the "In pursuit of" action?
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There are lots of options here, as is usually the case with this kind of question. As Sebastian's answer points out, a sequence of three datives (without et) is a good way to say "for X, Y, and Z", where "for" can have a broad range of meanings some of which are pretty close to "in pursuit of". Arguably more idiomatic than using abstract nouns (veritas) etc. would be to use the neuter forms of the corresponding adjectives, "true", "good", "beautiful"; this is a typically Latin way of talking about abstract qualities, just as in English you can say "the good" to mean "goodness". This would yield:
Vero, bono, pulchro.
If you want the "in pursuit" part to be more explicit, the most natural way to do this would be to put in a verb meaning something like "we pursue". But then the adjective forms will change because they become the accusative objects of the verb:
Verum, bonum, pulchrum sequimur / petimus.
(Sequimur and petimus both mean "we pursue, seek, aim at, strive for".)
The general rule for enumerations in Latin is that you have three options:
(I do not know where you got the spelling pulcritudo, it is at least non-standard.)
There are many ways to say “in pursuit,” but a pretty general term would be sequens. It captures the ambiguity found in the English (do you follow where truth etc. lead, or do you seek to attain truth etc.?). But in that case the three nouns are objects and go in the accusative:
Sequens veritatem et bonitatem et pulchritudinem.
But if your meaning is more clearly that you strive for truth, goodness and beauty, you could also use tendens, except in that case you also need an ad, but ad takes the accusative too, so that is all that changes:
Tendens ad veritatem et bonitatem et pulchritudinem.
Alternatively, in school mottos it is quite common to put such words in the dative case, meaning “for truth, for goodness, etc.” For example: Veritati et vitae (for truth and for life), Deo ac veritati (for God and truth); Virtuti, veritati, humanitati (for virtue, truth and humanity); and many many more.
In your case that would be:
Veritati ⋅ Bonitati ⋅ Pulchritudini
While this deviates a bit from your “in pursuit of” idea, I like this one best.