Hi @Anna_Dukhovich ,
This is one of the things that isn't sometimes that obvious when looking at Jira customization but I'll see if I can get you started here.
When you look at a Jira workflow, you see all the colorful status blocks with the little lines between them. The lines are actually where the magic happens. So... the short answer to your question is to set the assignee field as a post function to the "Create" transition. If you are in the workflow editor, if you click the "Create" line from the opening gray ball to the first step in the workflow, you will see the dialog pop up with a choice for Post Functions. Go there and then add the post function in.
The conditional part you want is also plenty doable. I would really recommend you add Jira Miscellaneous Workflow Extensions to your instance. That provides lots of ways to conditionally set fields including with Groovy script fragments (and it does a lovely job of leading you through the process). That way you could have the same workflow and, after testing the issue type, set a given human. You could either do this with an if, then, else Groovy script in one post function or have two; both of which set a different human for assignee but have it test in the conditions so it only sets that human for a given issue type.
Hope this gets you going. This page also may help to know where to go and what to do with it.
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