Learn Spanish (Spain): ¡Buen fin de semana! Gracias, igualmente.

The word bueno drops its final -o when it comes directly before a masculine singular noun. Since fin (end) is masculine and singular, bueno shortens to buen. You will see the exact same rule apply to words like uno (becoming un) and primero (becoming primer).

Where is the verb? The English translation says 'Have a', but I don't see that in Spanish.

In Spanish, it is very common to leave out the 'Have a' part for everyday wishes. While you could say a longer phrase that includes a verb, native speakers usually just say ¡Buen fin de semana! — literally 'Good weekend!'. It is short, friendly, and perfectly natural.

Because semana ends in an -a, is the phrase fin de semana masculine or feminine?

It is masculine, so you would say un buen fin de semana or el fin de semana. The main noun in this phrase is el fin (the end), which is masculine. The de semana (of the week) part just describes what kind of end it is, so it doesn't change the overall gender of the phrase.

How exactly does igualmente translate to 'you too'?

Igualmente literally translates to 'equally' or 'likewise'. It is the most common and natural way in Spanish to return a well-wish or a greeting to someone. You can use it anytime someone wishes you a good day, a good meal, or tells you it is nice to meet you.