First time I used the serial function I thought it was not working either, until I realized I needed to scroll down to see the letters I typed.
For RS-232 monitoring, I use the serial/state scope probe. With the 308 in default settings (neg logic, 8, N, 1,9600) it should just work. I connected a 9600 terminal to a breakout box and monitored pin 2. Set to HEX, press DATA= and then enter XX and XX for the data byte triggers. Press START and then start typing on the terminal. You should see the trigger light come on showing it is triggered and receiving data. Press STOP and look at the data. If you just see a lot of +'s then the data buffer is pointing to the wrong place in memory. Use SEARCH and enter the hex value of one of the letters you typed, or select CURSOR and use the position buttons to scroll the display. At some point you will see a highlighted CR on the left, binary in the middle and ASCII text on the right. The ASCII is shown in order going down the screen.
RS-232 is negative logic on the data pins so monitoring requires 0=neg for the data setting. If looking at the TTL side it may be positive logic so try the serial data setting set to 1=pos.
I did some experimentation and found that if you drive the Data Pod clock with a TTL square wave at 19.2K or 38.4K and set for Async EXT v (falling edge), you can monitor serial at 19.2K and 38.4K. It appears that the 308 will go faster than 9600 but does not have higher speed choices in the menu. It might be that it works ok with typed or keyboard repeat speed letters but not streaming characters (no character breaks). I did not try streaming characters.
The 308 is not a replacement for an HP 4951/4952, ARC or other data analyzer, which show characters real time, but will work in a capture function. You can tell if you are getting data and what the speed of data is.