2000-2006 Tahoe/Yukon was the GMT800 platform. 4.8/5.3 Gen III LS with NO cylinder deactivation. Transmission was a 4L60E. These are becoming harder to find clean, but otherwise very good trucks. Torsion bar suspension.
2007-2014 Is the next generation Tahoe, based on the GMT900 platform. 2007-2009 5.3 has DoD/AFM (Displacement on Demand/Active Fuel Management) cylinder-deactivation. 2010-2014 5.3 has DoD/AFM and VVT (variable valve timing). Transmission from 2007-2009 (although I've seen some 2009 with 6L80E) is a 4L60E, 2010 and up is a 6L80E.
Many schools of thought on this topic. I myself am looking for a 2007-2009 Tahoe with a 4L60E as I have one ready in the event of needing one. Some hate the 4L60E. Those are people who have had GM trucks with them, never do a fluid/filter change, and get mad when the trans is smoked at 150k HARD miles. If you maintain a 4L60E, they'll last. Plus in the event of a rebuild, they are cheap to rebuild. They have been around for forever and a half.
The 6L80E is a good trans as well. Better gear ratios, better for towing, clutch on clutch shifting. They can hold a lot of power. I cannot speak to their longevity, but I have friends with them in early 2010s Silverados who have yet to maintain them and they're at 120k+.
I don't really think 6L80E will have much MPG improvement over a 4L60E in a truck.
DoD can be deleted, there's many posts on here about it. You can disable it in HPTuners or with a handheld... if your AFM lifters and cam are in good shape. But if you have a lifter tick and or want to fully delete it -- Plan for a $1k at least in parts if you're going to do minimal. If you don't want to deal with DoD at all, look for a 2000-2006.
EDIT: ^+1, yes Clinebarger will have the total gospel and what to do. Fella is an absolute scholar when it comes to LS and GM trans.