Since you live in MA and work in RI, you would file RI a non resident RI return reporting only income that you earned IN the state of RI.
You would then file a MA return, reporting the income you received from all sources in all states, but you will take a credit for taxes paid to RI on your MA return.
To pay quarterly taxes, you can make the payments directly on the state and federal website.
To calculate, you will use your tax rate for that state and make the payment based on your profit for that quarter.
For MA, you can deduct the amount you owe to RI and pay the overage.
RI Tax payments
MA tax payments
Estimated Taxes: How to Determine What to Pay and When
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