According to the 2006 Chevy Trailblazer manual, the oil filter is located on the underside of the vehicle, beneath the engine.
Where is the oil filter on a 06 Trailblazer?
Finding the oil filter is harder than you'd think. In this case, it can be found on the base of the engine, towards the passenger side.
If you're interested in a obtaining the full manual for your Chevy Trailblazer, a PDF version of the document is available
here.
To get to the actual filter, you'll need to crawl underneath the car. There may be skid-plates blocking your access to the area, but you should be able to unscrew a small access panel directly above the filter.
If you're planning to change the filter yourself, I'd also do some research into the type of oil filter you'll need to order as a replacement.
One internet source quotes the AC Delco PF61 as the correct oil filter, although you may want to double-check that with your car dealer.