New to traveling

Welcome to the forum..
If your going with a 1500 GM product pulling a trailer be sure and get the NHT package (tow package). It comes with the higher rated gvwr and rawr numbers.

A 2014 and up 1500 gm you also can choose between the 5.3 engine at 355 hp/383 torque or the 6.2 engine at 420 hp/460 torque. Both can come with the NHT package. The 6.2 engine can tow GMs 1500 trucks tow ratings with ease.
Older engine were down on power levels.
And also be aware of some GMs engines AFM/valve train (cylinder deactivation issues) with the 5.3 and 6.2 engines prior to 2021-22 ?? models. Google has lots on this subject.

My wifes has a '16 1500 chevy 4wd crew cab short bed 5.3 engine 6 speed tranny 3.42 gears (not the NHT package). IMO a 6k-6.5k wet weight trailer would be the max I would want to pull around the country with the 5.3 engine.
The 6.2 engine can handle a 7k-7.5k lb trailer.

Good luck on the hunt.

Smaller trailer might be tough for family of five. Although kids might enjoy sleeping in a tent in the great outdoors. That will certainly help.

In today's economy, may be pretty tough to stay within 30-40K for truck and trailer. I don't want to say impossible, because someone will have an example of how they got truck and trailer for spit and a promise.

I would buy the best/newest/most reliable truck I could get. Last thing you want is to break down with family of 5 on side of the road. Make sure whatever you buy has a tow/haul mode and preferably selectable gear setting for the trans so you can have some non-brake control on the downhill side.

I would think a half ton with a 5-6K trailer could work out okay, but as JRScooby says, don't push it up hill, and take it easy coming down.

I will not tell you what you need, but you must understand even if you feed the biggest gas engine you will hear a lot of the mountains will slow you down
Point is if your load is to heavy you might not want to run WFO all the way up the hill to avoid overheating. As soon as the temps starts up, back out of it, let speed drop. Will others pass you? So what! A mile long climb, at 60 MPH, takes you a minute. Same hill, at 20 MPH, takes 3 minutes. Traveling the country you will find very few places that the biggest engine will run 60, and the smallest will pull down below 40. If the manufactures don't use bigger brakes with bigger engine the bigger engine not worth that much in mountains.

I have pulled a 5000lb 28 foot TT for many years with 3 different dodge rams. 2 were V8 1500s and a V10 2500 Ram. On the flatlands of Texas both my 98 and 01 Rams pulled the TT fully loaded both truck and ram without any problems at all and I used a Equalizer WDH.

I used my V10 2500 fully loaded to pull same trailer fully loaded thru the rockies and thru the west coast mountains of Oregon, California and Nevada.

My present 2014 Ram V8 hemi 1500 would also pull the same trailer in the west without any problems. You will need a bunkhouse mode TT for your size family or a heavier toy hauler TT that may be too much for a 1500 in the mountains.