There's like.. 8 different models of Silverados, but if you select yours on tirerack.com or other tire sites, it'll show you what the recommended sizes are. You can compare warranties, tread grades, etc.

I have a set of Goodyear wranglers on my Tahoe that are probably 4 years old with maybe 30k miles on them and are looking like this is the last winter. I thought they were identical to the previous tires that got 50k. I live in the same area, but go different places now, including some parking garages that probably wear them worse. But, they have been great for highway, off road (nothing crazy) and snow. And they're cheap (like $110 each). Not sure if I'm getting them again as 99.5% of my driving is on roads, but I would recommend them at least as a starting point.

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).