I'm not a fan of professional fitting since no one can "fit" the bike properly except the rider. The guy doing the fitting is likely going to fit you for his style of riding. There's plenty of fitting "rules" in the literature/on the Web to get you in the starting range.
Bouncing just means that one is not "spinning full circles" very well. It's a technique issue in IMO.
Concerning setting spinning goals, that might not be all that important. The key to my personal approach was to just keep the pressure on the pedals low, concentrate on spinning full circles (I.e. applying force to the pedal over 360 degrees, except on the upstroke where I prefer to just pull the knee towards the handlebars) and just spin to just below bouncing. Over time you'll find 110/120 is no problem. It's a mental or concentration exercise at first as is spinning full circles.
You'll also find the cadence where you feel best. For me it's cruising at 90 to 100. I spurt to 115+ on downhills. I have low gearing which helps spinning fast.
You really need a cyclometer with a cadence read-out if you're serious.
Al