I am not exactly a veteran, Thoraine, but it seems to me that quality confers a multiplier to all of the pets stats. While the difference between the respective multipliers is small (1.2 for uncommon vs 1.3 for rare) it will make a difference in the performance and survival of your pets.
That said, I have faced uncommon pets against my team of rares and had trouble beating them down, as things like breed, class and ability set matter much more than the difference between uncommon and rare. For more information read under Tools < Pet Guides < Guide to Pet Breeds.
I would say start out battling with one or more uncommon pets on your team just so you can get into it and begin the learning curve. Waiting until you have a large stable of rare pets is not needed, but you will see that it definitely gives you an advantage.
Last edited by Mwebly
on June 28th, 2013, 11:02 am, edited 2 times in total.