Normally, you file a state income tax return for the state of permanent residence (your domicile) that reports and pays tax on all your world-wide income, and file a non-resident return and pay tax in the state you are temporarily located that reports and pays tax only on income connected to that state. That would includes wages, or a fellowship paid to you while living and working in Illinois, but would not include investment income because that is assigned to your state of permanent residence. Since Washington does not have state income tax, you would only do the non-resident Illinois return. Since all your income is listed on your Federal return, you will need to manually allocate your income to Illinois when you enter the Illinois module in Turbotax (you will need to tell Turbotax which parts of your income should be taxable in Illinois and which parts should not).
Some states also have a "statutory resident" status where, if you stay in the state more than a certain number of days in a house or apartment (hotels don't count), you are a resident for tax purposes even though you are a permanent resident (domicile) of someplace else. However, Illinois does not appear to have that status. So you would be a non-resident and will pay Illinois income tax on your Illinois income but not your other income from outside Illinois.
