You need a digital storage oscilloscope (DSO) with sufficient memory depth to record however many characters you need to "see" at 9600 baud. If the data is serial ASCII, it typically consists of ten or eleven bits per character: start bit, eight data bits, one or two stop bits. If just one stop bit (the usual case) then 9600 baud means 960 characters per second. You need a 'scope that will take those 9600 bits per second, digitize them, and store them in memory. You should sample the waveform at about 100,000 samples per second to clearly discern each bit transition, and store perhaps ten characters of data, or as many as you think you need.
If the 'scope is sampling at 100 k per second, then each 10-bit character sent at 9600 bits per second will require 10/9600 = 1.014 ms. Sampling that interval at 100 k samples per second will yield (0.001014 sec) x (100,000 samples/sec) = (about) 100 samples. So about 1000 samples will be required to capture and display ten characters. Many DSOs exceed this memory depth, but if you need to capture longer strings of data you will need more memory. Or a digital logic analyzer which costs big bux.
I recommend you purchase
this inexpensive ($24) DSO available from Amazon and various e-Bay vendors. This is an easily assembled kit with all the SMD components already soldered to the circuit board. It only has 1024 memory storage locations, but that should be enough to get you started until you decide when, or if, you need to "move up" to a more capable DSO costing a few hundred dollars. The sky's the limit on DSO cost and capability however. You can generally expect DSOs with wider bandwidth and increased memory depth and multiple channels to cost more... sometimes a lot more once you leave the "sweet spot" hobbyist range of $200 to $300.
