How to choose a truck for camping?

Just start reading bud.
You'll see who has what riding on what.
Couple dinky pop ups (not 4 person campers) could go on a Taco.
Bigger pop ups can mostly be hauled on half tons.
Most hard side full feature campers will need a 3/4 ton at least.
Plenty of 4000-5000lb campers that are in 1 ton or dually territory.

Some people do just fine with a lightweight popup camper on a half ton truck, without a bathroom and a shower. Others need an F450 and three slideouts and a dry bath. There is no right or wrong, it all comes down to your personal needs and wants. You need to find out what is right *for you*.

Check out the buyers guide over at truckcampermagazine for an overview of current campers and their respective weights. Note that the stated weights often are not reliable.

If you have neither truck nor camper, find a suitable camper and get a matching truck.

As far as payload for campers goes, it is never a bad idea to have "too much truck". You will find that numbers that never meant anything to you as a car driver suddenly become meaningful, mostly GVWR (Gross Vehicle Weight Rating) and RAWR (Rear Axle Weight Rating). See also: Door Sticker. Tow ratings as well if you want to tow, but that is another topic that you need to research if/when it comes up.

A long camper with a lot of overhang on a short bed truck can badly overload the rear axle without overloading the vehicle. It will help if you understand what the Center of Gravity (CoG) on the camper means and where to find it.

It is of course possible to overload the truck, even though it is often illegal to do so. Many people do it without problems. Some things can be done safely, others not so much. If you don't know exactly what you are doing, just don't do it.

If the camper weight and truck capacity match well, the whole rig should handle just fine without modifications to the truck. The camper and truck in my signature handle very well together. But the truck is a fairly lightweight 2WD regular cab work truck.

As mentioned, campers can get very heavy, way too heavy for a Tacoma or other lightweight pickup truck. The smallest and lightest campers are popups but as you found they also get heavy. Don't forget about the weight of people, clothing, food, water, propane and all the other gear needed. I think you will find that a half ton truck is the minimum. If you want to carry 4 people you have another issue and might need a crew cab. That is more weight and also a small popup is not going to sleep 4 people comfortably or at all.

I looked at the Toyota Tacoma, but the specifications say 1120 lb maximum payload,

The 1120 lb number is a gvwr based payload which is spread over the trucks front and rear axles. However add 1120 lb in the bed may overload the rear axle/tires/suspension/wheels (RAWR).
I'll throw some payload number at you.

If your set on using a new model 1/2 ton truck to carry a truck camper then Fords F150HD has 4800 rawr which is good for around 2400-2500 lbs in the bed. That will include the fully loaded camper and all your "stuff" in the truck.

The new gen 3/4 ton trucks have 6000-6500 RAWR and are good for up to approx 3200 lb payload in the bed.

A one ton SRW may have around 3800 lbs payload in the bed.

And the one ton DRW with those big 9375-9800 RAWR can have around 6500 lb payload.
All numbers are approx numbers so payloads for any truck can be determined by actual scaled axle weights.

Or just use the truck makers GVWR based payload. Both are safe.

"... as we would like to go off road too ...". This stirs the pot, and requires more information. How do you define "off road"?. I have a modified 4x4 Samurai, that goes off road. I have a 4x4 DRW Dodge Quad Cab that goes off road. Hopefully it is obvious to you there is no way the Dodge can go off road where the Samurai does.

Tell us what kind of off road you have in mind, it will influence the truck and TC that will work for you.

Wayne

I would echo one of the comments above and start with the camper you want and then put the right truck under it. For the most part, the vast majority of people already have the truck so they are limited as to what camper they can safely put on it.
No knowing your situation well enough, it is hard to make a recommendation. Having said that, my recommendation would be to go with a one ton truck and that is only based on my personal experience. After that, you need to decide whether you want a short bed, long bed, crew cab, super cab, single rear wheel or dually.
You can see in my signature what my setup is. I use my truck as a daily driver and also for hunting, fishing etc. even when I do not have the camper on board. One thing about a one ton, I can always put on a smaller (meaning lighter) camper like a pop up if I wanted to at some point.

In my opinion I would forget about any truck in the 1/2T, 150/1500 class. If you are looking for a 2-4 person capacity camper then most small campers that will fit within a 150 truck will not provide the comfort for that many people.

Consider at a miniuminum a 250/2500 (except those with rear coil springs) and even consider stepping up to a 350/3500 truck.

You will need to consider more than just the posted weight of the DRY camper specs... water, food, camp chairs, firearms & ammo, fishing gear, bicycles, etc... whatever your rec activities dictate and add that weight to the truck carrying capacity... and it adds up quick!

If you want a truck camper just skip directly to a 3500 SRW as a minimum. You'll save a lot of time and money. I had a pop-up on a 1500 and it was overweight. Then I had a 11' hardside with no basement, no slide, and no kids on a 3500 SRW and I was overweight. Now I have a family and a dually I'm still overweight. Truck campers are HEAVY. Get a truck much bigger than you think you need.

Start with selecting/buying the truck camper (TC) you want. Then work backwards and focus on the right truck to 1) haul your TC 2) carry your people & possessions and 3) do the other things you want it to do i.e. off road.