Houston Texas to Montana

Stay on the interstate.

Livingston is 4,500 feet in elevation and near Bozeman Pass which is 5,700 feet. To enjoy your trip and see the great state of Montana you’ll need to do some mountain driving. No superhuman talents needed just a few very basic concepts that anyone can master. Read up and watch some YouTube videos. Experience comes from trying the inexperienced after study.

Interstate 35 to I80 to I25 to I90 to Livingston. We have traveled this route north from Kerrville area to Montana and northward to Alaska a couple times. No terrible climbs but it's uphill all the way. ELE 4501 at Livingston.

Interstates will be the least stressful. A lot of driving, so if you've got time built in, here are a few places to stop and stretch your legs:
Archway. I-80 Kearney NE

Little Bighorn Battlefield (if they've re-opened)

Cheyenne Welcome Center

I would like to suggest a different route which avoids Denver and is not so much a route that has Interstate driving. But it will take a little more time.

Take I-45 north to the Waxahachie exit which is Highway 287.
Follow 287 northwest through Fort Worth (or go around FW depending on the time of day).
Stay on north 287 in Colorado.
Near Limon take highway 71 north to Brush
In Brush switch over to Highway 34 through Greeley.
Then get on I-25 to I-90 and then on to Livingston.

You could also follow the first part of this route to Vernon, Texas. Then take 283 all the way up to Nebraska and then take Interstate 80.

I enjoy the US and state highways.

These routes are pretty flat except in Montana and you will avoid the traffic and tolls near Denver.

X2 for GizmosMom's post.