You really aren't giving us enough information to realistically tell you what to do. I think the simplest thing for you to do is to go for a kitbot with mecanum wheels, make a basic frame to mount these (
http://www.andymark.com/product-p/am-0482.htm) and just put mecanum wheels on the output shaft or just buy one of the mecanum platforms andymark already sells.
Q:
What is the suggested RPM for Mecanum wheels?
A:
what size wheel do you have and how fast do you want to go?
Q:
What motor has had the best outcome?
A:
Not enough information. Is this FRC specific? What limitations are you working with in your class? Most teams in FRC use 1 CIM motor per wheel but other teams have gone for 1 CIM + 1 miniCIM per wheel as well. I will withhold my opinion on the competitive outcome of mecanum based robots in FRC other than that the faster ones with better drivers are visibly better.
Q:
Which is best for motor reduction? (ex: gearbox, chain, belt, etc)
A:
Not enough information. This depends on your implementation and design. This year, on their respective tankdrives, 254 had all gear reduction, 1114 had gear+chain reduction and 67 had gear+belt reduction. Most mecanum drives I see are all gear.
Q:
Any other helpful tips about building with mecanum wheels?
A:
Excessive rigidity is bad. Uneven weight distribution is also bad. You want all wheels to have equal traction at all times. Unless the floor is flat and fairly smooth I'd avoid using mecanum wheels.
What budget and manufacturing resources do you have access to and what surfaces are the bot going to be driving on?