The biggest problem with ASM admin role is that you don't have access to the command line, so you cannot get a full ASM qkview. The qkview you get from the TMUI (web page) does not include the ASM Event logs, which are almost always required to help with troubleshooting. So each time you open a case with support you will have to ask the full admins to run either
asmqkview --add-proxy-log (10.2.2- 11.5.x)
or
asmqkview --add-request-log (11.6.0 and above)
Also you will need to connect with them if you need to run a tcpdump.
Note that you can get around this partially by exporting the relevant event log as a PDF and uploading to dropbox.f5.com.
As the primary troubleshooter you will want to start with your event logs, as they are likely to tell you exactly what the problem is. Another useful tool is httpwatch. Some people like fiddler, but as a browser plugin httpwatch shows exactly what the browser is seeing, rather than acting as a man in the middle attack.
If the problem is ASM, rather than traffic you will want to look at /var/log/asm and all the files in /var/log/ts/. Note that this is a lot of data and can be confusing. Support can help here.
You can also check the policy logs, which can tell you who made changes to the policy. These are going to be the places you will want to start.