Re the 3 jaw chuck I have been well served by a 6'' Phase II version, it is a budget chuck that represents great value.
Bear in mind that tooling up now without having a clear notion of the work you will be doing is risky. You might end up with tools you don't use. Also, if later down the line you want to upgrade to a bigger lathe much of the tooling for your first lathe will probably not work on the new one.
At the hobby level we can take a few years to accumulate the tools we want. Often, it is rewarding in itself to use what we have to the limit and find ways to make do with what we have. Then it becomes more clear what our next tool addition should be.