Yes, there is a very good chance that your ABS and TRAC lights are coming on because of a defective wheel speed sensor. There are also many other things that can cause that including a failed traction control module or ABS hydraulic control unit.
Diagnosing the exact cause of the problem is not really a DIY job. You will have to access the computer data and retrieve the fault codes from the ABS System. Once the codes are collected, the problem must be diagnosed according to the manufacturer's troubleshooting charts for each code.
Changing a defective part might be a candidate for a DIY project, but that depends a lot on what is actually causing your problem. Sometimes these things are fixed by reprogramming the computer - no hard parts actually get replaced in these cases.
Even if it is a wheel speed sensor, there are sensors at each wheel on most cars. (You didn't mention MAKE, MODEL, ENGINE SIZE, TRANSMISSION TYPE, Etc.) In this scenario, you will have to access the computer data just to find out which one it is.
The hand brake thing could be anything from a broken cable to a simple adjustment. Again, the
CAUSE must be diagnosed.
Please also review this article:
What Else Could Be Wrong?