Your AT will do anything a KLR can do, and probably better...
I put 17K miles on my KLR and I had a blast doing it. Bang for the buck, it is a hard bike to beat, especially if you pick up a used one that has already had all the necessary upgrades/mods done to it. I am not talking about performance mods. I am talking about just stuff that fixes some glaring short comings. The KLR is inexpensive for a reason. It is built with lower quality components.
First, you need to do the Doohickey. Google it. If it breaks before you upgrade it, it can nuke the motor. Some people get lucky and it never breaks. Others... not so lucky.
Second, subframe bolt kit upgrade. MANY of the bolts used to hold the bike together are weak. It is common for the foot peg bolts to shear off or for the threads into which they screw to strip out. So you have to heli-coil the threads and get new bolts. It you mount a center stand and it takes a hit (like a bash plate would), it slices the bolts like a knife through hot butter... Been there, done that. The bolts that hold the subframe to the main frame and which support your weight and that of any luggage are also weak. It is common to see those fail under load. The upgrade kits have high strength stainless bolts for replacing the weak ones. Mine came apart while riding in the Gila Desert East of Phoenix and we spent the better part of the next morning cruising to different auto part stores and Action sports stores looking for parts so we could fix it.
Third, the suspension just plain sucks, bad. Stock suspension is really soft and the front end of the bike dives like crazy even under moderate braking. There is NO adjustment on the front suspension and only preload/compression on the rear. I had to put in stiffer fork springs, slightly heavier oil, and change the preload spacer to get the sag right. Some people go as far as replacing the damping controls because the stock ones are so harsh and non adjustable. I had to put a stiffer spring on the rear shock as well. For reference, I weigh about 180-185 butt naked, maybe 200 with all my riding gear. If you don't upgrade the rear spring and/or do a fenderectomy there is a good chance you'll experience an unplanned fenderectomy when the back end bottoms out and the fender catches on the rear tire...
Fourth, the brakes suck. If you start looking, you fill find that people have done all kinds of mods trying to improve the brakes. The easiest is simply stainless brake lines and better pads. That will make them better, but they will still suck compared to many other bikes.
Fifth, the motors are jetted really lean from the factory to meet emissions standards. Google the two cent mod. That is just the start of trying to get the engine running better. Many people put JD Jet kits in them.
Sixth, the stock gearing is a bit tall if you want to do anything like the K-Trail. This make you have to lug the engine at low RPMs in first gear. A common solution is to go one tooth smaller (13 teeth) on the front sprocket. This gives you about another 500 RPM, which makes hill climbs and descents MUCH easier!
Seventh, there is not much extra power for running accessories. To be fair though, this is true of many carbed dual sport/dirt bikes. They just didn't need a lot of power as they expected to only be running a headlight, brake light, and signals. They weren't designed with all the electronic goodies, heated grips, heated gear, etc,... in mind. The more recent FI bikes generally have more electrical power.
All that said, the bike is a great cult bike. They have been very popular with world travelers because they are reliable and simple. They are easy to work on and don't require much in the way of special tools. You don't need any kind of diagnostic computer gadgets to view codes as is the case with many FI bikes now. They don't do much of anything really well, but they will do a lot moderately well. They come with the big gas tank and decent seat compared to other bikes.
There are countless threads on AdvRider, various KLR specific sites, and even here on TWT where people are trying to do everything they can to "fix" the bike and make it better. You CAN spend a good bit of money doing that. However, it will still be a top heavy KLR. If all you do is ride street and basic county/forest roads with it, you will be happy. If you start trying to do stuff like the K-Trail, then all those shortcomings will start to rear their heads and demand your attention, attention that is better spent focusing on the riding rather than compensating for the bike's lack of ability. There ARE riders out there that make KLRs do amazing things, which is true of pretty much all bikes, but most of us are not that level of rider.
As was mentioned above, it is VERY common for riders to start on a big bike, like a BMW GS, V-Strom 1K, etc,... and then transition to a KLR... then to something else. The thing about the KLR though is that, for whatever reason, they make you bond with them. I think it is the shared experience of suffering through things together
My KLR and I did a lot of that kind of bonding
I picked up my slightly used KTM 530 EXC for about $5500. I think it had 1700 miles on it. The previous owner had added some nice farkles. Every time I ride that bike, especially if things start getting technical, I remember how hard I had to work when riding the KLR and all I can think of is how glad I am that I am not riding my old KLR. This is REALLY true when I have to pick up the bike!! Picking up the KTM is easy, even if loaded. The KLR was WORK and I often needed help. The difference in handling, power, brakes, weight, and pretty much everything other than seat comfort is incredible. KTM needs to hire a new seat designer... I guess they expect you to be standing all the time!
I agree with the other folks that say hang on to the 250 for now. Upgrade the springs, which is not expensive. Then just use that bike to really focus on developing specific riding skills. I think you will find, like so many others, that you will tend to be looking for challenging and technical riding more often than you currently think you will. Even if you don't, you will still be better at handling the unexpected challenges. Part of adventure riding is encountering the unexpected and being able to deal with it. Good skills go a LONG way in that regard! Once you have more seat time and more adventures under your belt, you will have a MUCH better idea of what you really want to do and what you need to do it. If you really only want to do the mild stuff, there's no reason to get a different bike. The AT 1000 will do all that with ease and comfort.