The important thing with any type of dishwashing scrubs/sponges etc. is to work hard at figuring out whether they've had any type of soap or cleaner impregnated in them. Its surprising these days how many products these days will have this but not indicate this on the packaging. You have to be a detective: study the packaging closely, use feel and smell and satisfy yourself that you think its just plastic (or whatever) and does not have soap. At home, put it in some water and test it again with touch, smell and perhaps taste.
Once you -have- found a nice plain plastic pot scrubby or whatever, note the brand etc. so you can get them again hopefully when you need them. (Soap can kill fish!)
Any new tank or items should be rinsed with fresh tap water and scrubbed a bit, depending on need. It sounds like this is all you'll need. If the white spots turn out to be tough mineral deposits then yes, vinagar might help. The usual sequence we recommend with mineral deposits would be sponge with a little vinegar, followed by plastic pot scrubber alternated with rubs with a lemon/lime rind that still has some of the fruit (don't get a lot of either of the these acids on the tank joints) and finally the use of a single-edge razor or similar, alternated with the citrus rind etc. Alway do thorough final tap water rinses.
~~waterdrop~~