I've done track days in my 01 off and on for the last 20 years. I've never had any issues with trans cooling, but I agree that brakes are a problem. I tried everything I could think of to make the stock brakes work and never had much success with that. My first few track days cost me $1000 in brake parts every time I went to one. Then I tried a few different iterations of Wilwood brakes before settling on my current set up. I got all my stuff from TCE and currently have their 14" front kit and 13" rear kit. I've had no further brake problems and the support from TCE is awesome.
Once you get the brakes up to speed there's a lot of ways to head. If you want to focus on durability and longevity, you may look at shocks next. There were a few times I felt like the stock shocks were losing control towards the end of a session in the summer, so those probably need upgraded or at least some kind of cooling added to them. I run QA1 adjustables from RUSlow, but Stan closed his shop a long time ago and I'm not sure if the shocks are still available direct from QA1 or not. There may be other options today as well.
Another option is to improve the handling. This doesn't necessarily mean cornering grip, but more how the truck feels on the track. For that the best thing you can do is a Panhard bar, especially if you have lowering shackles. Really tightens up the rear end. Next I would upgrade the springs. I'm not sure what is available now. I run RUSlow 1100 lb/in front coils and old style Hotchkis rear leafs. I'm not sure how much that lowers the truck, but I'd guess an inch in front and 3" in the back. I also have the Hotchkis sway bars and with those the rear leafs are a little too stiff. If I was doing it again I would probably go with the new style leafs that are a little softer.
If you are on a tight track the stock Traction Lock will quickly show you how useless it is. I put a TruTrak in mine and it puts the power down coming of the corners a lot better.
Last spring I threw the entire Stiffler's catalog at it. Frame stiffener, trans crossmember, traction bars, driveshaft loop, and front suspension arms. Well worth it. Really tightened up the whole truck and I highly recommend their stuff. The only thing I didn't like was the hardware that came with the upper control arms. It was too short and put the load in the threaded part of the bolt. I asked them about it and don't believe they even realized why that was a problem. I ended up just getting longer bolts that allowed a smooth shank to carry the load and went about my business.
Lastly, tires. I mention this last because unless you've got most of the above the tires won't really help you and will simply get you to the limits of everything else more quickly. If you're on stock wheels the only real option you have is the Nitto NT-05. They are great tires, but they are pretty outdated and good luck finding any. Took me 6 months to locate a whole set. If you are replacing your wheels I strongly recommend going to either 17s or 20s to give you more tire choices.
ETA: I meant to circle back to this and forgot. I don't have any significant engine mods and that may be the reason I don't have trans cooling issues. So far all I've got are long tube headers and full exhaust. I also have the Ford 4x4 trans pan and run Amsoil fluid. With the stock pan (it holds a quart less than the 4x4 pan), regular fluid, and more power to hold this could be an issue. I just got a 4lb pulley and tune to install in the next few weeks, so I guess I'll know at my next track day how well the trans handles that.