Learn Turkish: Müsait bir gün bulalım ve festivale katılalım. - Let's find an available day and join the festival.

It comes from the verb root bul- (to find). To say "let's do something" (the optative mood), we add -a or -e depending on vowel harmony, and then the "we" ending -lım. Since bul- has a back vowel (u), we add -a to get bulalım (let's find).

Why is it müsait bir gün instead of bir müsait gün?

In Turkish, when describing a noun with both an adjective (like müsait, meaning available or suitable) and the word bir (a/an), the adjective comes first. The standard order is always Adjective + bir + Noun.

Why is it festivale and not festivala? Doesn't the last vowel 'a' require a back vowel suffix?

Normally, yes. However, festival is a borrowed foreign word ending in a "soft" L sound. Because of this, it breaks the usual vowel harmony rules and takes the front vowel -e for the dative case. Note that the dative case (meaning "to" or "towards") is required here because the verb katılmak (to join) always demands it for the thing you are joining.

What is the root of the second verb, katılalım?