Vinegar is the safest route for getting rid of calcium deposits and other mineral based stains. Warm water and a scrubbing pad should get rid of dirt, dust, etc. If the vinegar isnt working, try using a razer blade on them after letting the vinegar soak a little.
A "cleaner" but riskier method is to dilute bleach in water, ~5% bleach, and use that to scrub. However, you will need to really rinse the tank thoroughly. This method is usually not advisable unless you really need to tank to be spotless.
As DKRST said, most stains will fade after the tank is filled, and depending on water conditions, might even dissolve on their own after a while, soft/acidic water will eat at mineral deposits slowly (by slowly, i mean can take years). But light stains should not be noticeable for the most part.
As for the filter, i would give it a good scrubbing with the hottest water you can handle. No soap/chemicals of any kind. If its an HOB, I would invest in a pipe cleaner and clean out the intake tube. For canister filters, buy new tubes. You will also need to locate the impeller and clean it, dried mulm, algae, and other solids could damage it when you start it up again.