Trenitalia site doesn't seem to work for this route
Well, it does, but offers only full fare tickets which are kind of expensive, so might be better, as you mentioned, to do a 'break' via Verona. Then you have a chance to buy two discounted tickets (19 euro each), while next just buy discounted one-way 'EUROPA 1' tickets:
For return leg (Innsbruck-Verona) you also can buy 'EUROPA 1' via TrenItalia or same discounted tickets, but from OeBB website. As result you pay 58*2 for Verona-Inn.-Verona. Although remember that both 'MINI' offer 19 euro discounted tickets (Florence-Verona) as well as 'EUROPA 1' Verona-Innsbruck are not exchangeable or refundable, because both are promo offers sold in limited number per train.
For Verona-Bergamo it does make sense not to book anything in advance, but just use regional trains and thus pay only 9-10 euro without any ticket prepurchase
I also found a site called 'raileurope.com' ... Does it work for me? Can I trust it?
This is an agency, which basically re-sells rail tickets with own commission, so try to avoid such middle-man, both in order not to overpay (especially when same ticket can be bought from official rail company website) and stay on a safe side (--> not deal with a agency website)