First off, make sure the UPS works with the Combine Shipping DISABLED. If the UPS isn't working without combine shipping, it isn't going to work with it. (Getting UPS, USPS, FedEx, etc. working in the first place can sometimes be rather "fun"...)
As far as the flat rate goes, if you want to specify the flat rate at the product level, make sure the "/admin/store/settings/quotes/methods/flatrate" Flatrate method is set to 0.00 for both the base price & the default price.
(Base price = starting price for all flatrate, if the Base is 5.00 and you tell a product to have a 10.00 flat fee, the total at checkout will be 15.00)
(Default Price = If the default price is 10.00, all products will inherit the flatrate shipping cost of 10.00 unless it is changed at the product level)
If you already knew this, sorry... You mentioned drupal / ubercart being a new experience...
I'm also assuming you went into the "Shipping Method" field for product content type and specified the key names for the flat rate options you have enabled? It gives I think two default flat rates out of the box... I usually just delete those and create two of my own called:
Flat Rate Shipping
Those make a more sense to me than the default ones.