How do you know your phone has rebooted? Just because the lockscreen appears doesn't mean it rebooted. Check your battery stats, Settings > About Phone > Batter and look at the up and awake time. If the phone has been up for a long time its unlikely your phone is rebooting on you. Just check this after a time when you think the phone has rebooted.
I think you are going to have to use logcat to get logs from an Android phone. If your phone is rebooting on you, you could attempt to run logcat, but you would have to have logcat running continuously to capture what is going on and making it reboot. This is not very practical. But look below for info on logcat.
Instead of using logcat I would try just starting from a fresh install of Android. If you are rooted I would backup, wipe, and install a ROM from scratch. If you are not rooted, I would download the latest version of Android for your phone from HTC and install it. This will wipe your phone, but running from a clean slate should fix the reboot issue.
Once on your fresh Android install open Market place and install your apps starting with essentials only. DO NOT INSTALL any task killer of any kind. Reboot and see how the phone runs for a few days. If things seem fine then move on and install your other apps a few at a time. Give it a few days and continue on to the next few apps until you find the app that is giving you problems or maybe all you needed was a fresh start.
Install the Android SDK on a computer and then open a terminal shell (cmd.exe in Windows) and run adb logcat. On windows it looks like:
Sometimes I have to specify the whole path for that command to work. For me it would be:
You mileage will vary depending on where you install the SDK...
After running that command you will only be able to view what the terminal shell can contain or hold in memory. My recommendation would be to increase the buffer size of the terminal shell so that you can capture more information. More advanced shells might be able to output a text file, I would try that too. If you do not have this option just copy and paste everything in the screen to a text file yourself. In cmd you do this by right clicking, mark, highlight what you want, and then go to the text file and paste.
Also this is not always the best route unless you know what you are looking for because logcat will essentially show output of everything your phone is doing. So there will be a ton of data generated.