So assuming that this is not some odd grading system. Their oil change guy told you that there is 3/32 of tread on your rear tires. Tires usually have about 10/32 new and are worn out at 2/32.

Basically he was telling you that you need new tires. Though it is always good to check this stuff yourself. As he would probably get a cut of any upsale. It doesn't hurt to get a tire depth gauge and measure the area of the tread that has the lowest tread depth.

As for actual tire brand recommendations. I have had very good luck with Cooper Tire's white label brands. Which are Dean Tire, Mastercraft Tire, and the Big O Tire House brand(they may have others too). They make a very nice all season tire and they have done well in the winter for me. They are all made in the USA at Cooper's tire plant where they also make Cooper brand tires. As far as I can tell the difference between the white label tires and the Copper ones is they are using the last version of the Cooper Discover tire molds and they don't put extra kevlar plies in them (you don't need that unless you are going over a lot of sharp rocks).