How often do you wish to change your oil? Less than 5000 miles? 10W-30 Chevron, Mobil, Pennzoil, or Castrol will work very well. Also working well are the over-the-counter synthetics (excluding the Mobil 1) and blends, but it's hard to justify their cost compared to the service they give.
Up to the max oil change mileage allowed under your warranty? Any of the true synthetics, 10W-30 Mobil 1, Royal Purple, Amsoil (except XL) or Schaeffer synthetic blend (the best value).
Extended oil drains combined with perodic oil analysis?...any of the above, especiall Amsoil.
Filters...ACDelco, WIX & NAPA/GOLD, Purolator, Baldwin/Hastings, Bosch Premium, are good, and Wal*Mart's SuperTech isn't bad.
I have a 2003 Silverado and we probably have the same engine...5.3? I have been using Mobil 1 5w30 and I will soon have an UOA done on it to see how it is working. So far the engine seems to like it! After reading this forum for awhile it seem that the WIX filters like Napa Gold 1522 and Car quest preimum are the best for these Vortec engines.
HC, since it's a 2003 model I assume it'll still be under warranty for a while. If it were mine, I'd use AC-Delco oil filters. Should a warranty problem pop up, that removes a possible point of contention right there. How can a GM dealer tell you that your problem was caused by a good ole AC filter?
Otherwise, Wix, Hastings, Baldwin are all good choices.
For oil, I'd make sure it's a good API certified SL oil, in whatever weight the owner's manual recommends. I'm happy with BobZoil, but there are other good choices. For OTS dino, PennZoil, Castrol, Chevron, Mobil Drive Clean & some others all tend to produce good analysis reports. OTS synthetic, that's pretty muchlimited to Mobil 1 now. Then there's Amsoil, & a couple of others.
Once the warranty is up, I'd run whatever filter & oil I liked best!
If you can find the right Delco at a reasonable price, go with it. I have found the PF 1177 has half again the filter area of others. STP, Hastings, Purolator, ST, etc. were in a group with twice the area of a FRAM.