Lots of Options
There are as many answers to this question as there are people.
For me, the shortest answer is a Toyota Land Cruiser, that can burn high sulphur fuel, with 750x16 tube tires and a roof top tent. Note that you probably can't buy this vehicle in the US.
I wouldn't consider a trailer and, with age and experience my minimums are:
-- Indoor sleeping/cooking
-- Nothing that needs to peg into or attach to the ground. (Some times you have to "camp" in a town square or at a border post.)
-- 500 mile range
-- Food/water for a week
The best, most academic study of this issue may be here:
http://www.xor.org.uk/silkroute/equipment/choosevan.htm (But notice that in the end, he ended up buying a used Bimobil!)
But there are variables:
-- Why are you doing this? Do you really want/need to go "Around the World?" Consider making several, shorter trips.
-- Comfort matters. I have met a lot of burned out overlanders - tents, customs, police, language - after a month or three it can get to be too much. You stop traveling and start simply surviving.
Some suggestions:
-- Start following the blogs/web pages of people who are really doing this. Add a tremendous BS filter and learn to catch the clues of some of the really expensive problems they have had. For various reasons, most bloggers want to make it sound really easy.
-- Unless you are already experienced campers/outdoors people, try your hand at some shorter trips near home. How do you react to downpours or snow? This may tell you that you want a hard sided camper.
-- How much have you traveled overseas and how many languages do you speak? This will give you some sense of how well you can deal with foreigners for extended periods of time. (Shocking, but the world is full of "foreigners.")
-- What are your mechanical skills and inclinations? I had Chevrolets in Africa for decades, but I did, or supervised, all of my own work and I had the specialty tools needed. Less handy? Get a Toyota.
This web site is a gold mine of information and links to people who are doing just what you want to do - learn from their experiences.
And have a great, safe trip.