Can't help you on best time to go although some folks say nearer the end of the day versus the beginning. Whatever you do, order your tickets in advance!!!!!
http://www.b-ticket.com/b-ticket/uffizi/default_eng.aspx
Unfortunately, reduced ticket prices for both Uffizi and Accademia are only available for citizens of the E.U. ages of 18 to 25. From the B-ticket website:
"Fifty percent reduction for European Union citizens aged 18 to 25. Proof of eligibility for reduced-price tickets must be presented when picking up your tickets.
So, Americans over the age of 17 pay the full adult ticket price.
The duomo itself is free but there's an entry fee for the other parts of the complex (dome, museum, bell tower, etc.) Tickets include entry to all of those parts, and discounts only apply to children ages 6-11 (plus some others you don't qualify for). Reservations for the dome are MANDATORY in advance so if you want to climb it, do that when you buy your tickets online:
https://grandemuseodelduomo.waf.it/en/cumulative-ticket-great-museum-of-the-cathedral-of-florence
If you have a full three days (or 72 hours) in Florence and want to see enough of what if covers to break even on it, you might also consider the Firenze Card:
http://www.firenzecard.it/index.php?lang=en
One caveat: you must still make pre-reservations to climb the duomo's dome if you have the card, and you can't use the duomo's ticketing site to do that: you have to wait until you arrive in Florence (see the Firenze Card FAQ page). You will stand less of a chance of landing a time slot than you would if buying tickets/making reservations through the basilica's own ticketing site. Climbing the bell tower instead can be a good substitute as demand for that one is not as high. If climbing neither of them is important to you, then no problem. :O)
