What Duffster said is correct. Generally the level at which you need a CDL is 26,001 LBS. Being "NOT FOR HIRE" or hauling "just my stuff" is irrelavent. The exception to this is the "10,000 LB trailer rule". Yes you can have a trailer up to 10,000 LBS behind a 26,000 LB truck and you don't need a CDL. The problem is most trailers pulled behind a 26,000 LB truck will have a GVWR higher than 10,000 LB.
Here's the problem that get people in trouble. It's not just the GVWR of the vehicle that is looked at. You also need to look at ACTUAL WEIGHT and REGISTERED WEIGHT. If you have a trailer that has a 14,000 LB GVWR and then register it at 10,000 because you are trying to get the 10,000 LB trailer rule to work for you...it won't. If ANY of the three (GVWR, actual weight, registered weight) are over, you need a CDL. If you have a dump truck registered at 26,000 LB and the GVWR is 26,000 LB but overload it and now it weighs 29,660 LBS...CDL REQUIRED. Basically what I'm saying be careful and look at more than just the GVWR.
There are cases when the power unit is a 3/4 ton pickup and you need a class 'A' CDL. A 3/4 ton truck with a diesel normally has a GVWR around 9,000 LBS. Put a goose-neck trailer behind it with a 20,000 LB GVWR and your at a total GVWR of 29,000+.
Another issue not discussed here is US DOT numbers. If the unit is over 10,000 LBS and is "in commerce" or otherwise it's a business (even a"side business"), they need a US DOT number.
Aren't you mincing intrastate/WI regulations with interstate/Federal regulations?