Just to clarify a minor thing, SQL Server doesn't usually create timeouts, rather the calling client is what creates the timeout when it exceeds its wait time threshold while waiting for a response from SQL Server. This threshold is configured by the calling client itself, and can vary from one application to another. The only exception to this, is when SQL Server is accessing data remotely across a Linked Server. Then SQL Server is the calling client in that case, and can issue a timeout if you configured your Linked Server to have a timeout threshold.
That being said, it sounds like you're just looking for general ways to debug long running query issues, especially with step one of identifying what those queries are and why they are running slowly.
One of the best tools for this, in my opinion, is Adam Machanic's sp_WhoIsActive. This procedure tells you every query running on your server at any point in time, with information on how long it's been running for, if it's being blocked by another query, what the last wait type was, what the CPU usage is, the query text itself, and more.
This is basically my first step whenever I need to look into slow running queries. I actually log the results of this procedure every 10 seconds to a Table, so I can retroactively look into performance issues as well.
Another awesome and free tool is Brent Ozar's sp_Blitz (and the correlating "Blitz" procedure suite) which gives a lot more in depth information on potential issues with your server, your queries, and any configuration issues. This is a more advanced tool, so I'll use it secondarily for answering the question "why is a query running slowly?" after using the aforementioned sp_WhoIsActive.
And finally, a third tool I'll recommend that I've recently added to my arsenal, is Erik Darling's sp_PressureDetector. This tool is great for quickly identifying CPU and Memory pressure issues on the spot. This provides more high level information which can help you determine if your server is under-provisioned or if you're having serious issues with the CPU or Memory which can correlate to query performance problems.